
THOMAS HILL D.D. 



LIBRARY OF CONGRESS. 



ail. fiXeajtiirigi "^txHSl 
Shelf. 






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UNITED STATES OF AltfERICA' 



GEOMETRY AND FAITH. 



"No man, therefore, can doubt, but toward the atteining of knowledg 
incomparable, and Heavenly wisdome, mathematical speculations, both 
of Numbers and Magnitudes, are means, aids and guides; ready, certain 
and necessary. " — JoJm Dee, of London y 1570. 



GEOMETRY AND FAITH 



A SUPPLEMENT 



NINTH BRIDGEWATER TREATISE 



THOMAS HILL 



" The truths of Natural Religion are impressed in indelible characters 
on every fragment of the material world '* 



THIRD EDITION GREATLY ENLARGED 






BOSTON 
LEE AND SHEPARD PUBLISHERS 

NEW YORK CHARLES T. DILLINGHAM 
1882 






^S^' 



** This is the chiefe Glory of Geometry, that it loyters not, or employes 
it self about these inferiour Machines, from whence it had its Original, 
but hath soared up into Heaven, and resetled humane minds, (groveling 
before in the dust) in CiXJlsstial Seats." — John Leeke and George Serle, 
1661. 



/l-d^Of\ 



Copyright, 1882, 
By lee and SHEPARD. 



All rights reserved. 



fc 



CONTENTS. 

CD 

^ Page. 

4 CHAP. I. SYMMETRY IN SPACE i 

( II. SYMMETRY IN TIME lo 

III. NUMBER 15 

IV. THE CALCULUS 20 

V. APPLIED MATHEMATICS 24 

VI. MOTION 27 

VII. MUSCULAR ACTION 32 

VIII. GEOMETRICAL INSTINCTS .... 37 

IX. MOTION ETERNAL IN DURATION ... 43 

X. MOTION OMNIPRESENT IN SPACE . . 46 

XL THE SPHERE OF HUMAN INFLUENCE . . 50 

XII. MAGNITUDE 54 

XIII. CHANCE AND AVERAGE 64 

XIV. PHYLLOTAXIS 7^ 

XV. NUMBER AND PROPORTION .... 90 

XVI. THE DEVELOPMENT OF FORMS . . 102 



* * * y^coiiergiav^ * * o dfj Oav^a ova dvdQcimvov, alia yeyovbg 
deTov (pavEQOv dv yiyvoiro r^ dvvafisvco ^vvvoieiv' * * o dt delov r earl 
xal dav^aoxov roJg syKado^jwai te aai diavoovfisvoig' — The Epinomis, 



PREFACE, 



"The present work lays no claim to originality." A humble 
gleaner in the fields of Mathesis and Theology, I offer only a few 
of the common fruits, well known to those who have more thoroughly 
surveyed the boundaries of these two domains. And I have ventured 
to connect them with the name of the " Bridgewater Treatises," not 
because I consider myself worthy to appear in company with their 
writers, but simply that I may thus the more earnestly express my 
admiration for the treatise of Babbage, 

Waltham, Mass., 1849. 



PREFACE TO THE SECOND EDITION. 

I have rewritten the greater part of this work, and altered so much 
the expression and illustration of my thought, that I might have given 
it a new title, but for the affection which twenty-five years' familiarity 
has bred for the old one. 

Portland, Me., 1874. 



Eviiokoyrp^ov, tcpij* rov yag dai ovrog ?/ yemfittQtK^ yvc^aig laxiv 
'Olxbv ccQa, w vsvvais^ V^y^^g TZQog dlrideiav eirj av, xal anegyaanxov 
cpiXooocpov diavolag ngog to dvco a)(eiv d vw xdzo) 6v dsop e^^Ofitv 
— Plato's Republic, Book VII, 



GEOMETRY AND FAITH. 



I. 



SYMMETRY IN SPACE. 

The universe, actual, possible and impossible, is composed of 
four elements, spirit, matter, space, and time, which are by no 
alchemy transmutable into each other. Many alchemists continue, 
even in this closing half of the nineteenth century, to make the 
attempt, and some even flatter themselves that they are succeed- 
ing ; but the sturdy reply of human consciousness is, that the four 
elements are diverse and not transmutable ; or, if any trans- 
mutation is possible, it must be confined to this, that matter may, 
in some manner, be an effect of spirit. But to us, finite spirits, 
nothing more is granted than the re-arrangement, the partial con- 
trol of matter, not its creation. Matter, as we know it, is dis- 
tinguished by its being the recipient and dispenser of force ; which 
force, so far as we know it, is from spirit alone. This obedience 
of matter to spirit gives justification to our suspicion that it is the 
creation of spirit. 

Space and time are without parts, and are indivisible except by 
a mental act. This division is suggested to us by manifest motion 
in matter. Force shows itself in matter by moving it ; that motion 
calls our attention to the space and time, within which the motion 

(I) 



2 GEOMETRY AND FAITH. 

is taking place ; and we divide mentally this space and time, first 
from the remainder of the boundless contiguities, secondly into 
smaller parts. Thus geometry and algebra are generated, the 
sciences which deal respectively with space and time, those pure 
entities, the relation of which to the Infinite Spirit we cannot com- 
prehend, but w^hich we become familiar with in the finite portion 
embraced in our experience, in the universe and its history. 

In geometry, the mind imposes upon indivisible space arbi- 
trary boundaries of division, according to arbitrarily selected laws 
or conditions. These boundaries are of three kinds, surfaces, lines 
and points. The point is a zero of magnitude in space, but never- 
theless is not nothing ; which is nowhere, while the point is some- 
where. This contradiction in terms, that a point should have no 
extension, and yet have a position, is one of those instances, in 
which geometry abounds, in which the mind is compelled, by the 
necessity of direct vision, to admit each of two truths, which are to 
logic mutual contradictories. The mathematician modifies the law 
of non-contradiction by confining it to propositions concerning finite 
quantities. 

A lower form of a zero of magnitude in space is the hne, which 
'tended, at each point, only in two opposite directions ; and the 
it form is the surface ; for which there can, at each point, be 
1 a line, such that the surface extends, in every direction, 
perpendicular to that line. Geometers define these lower 
s of zeroes, or boundaries in space, by the further self-contra- 
on of imagining the movement of a point ; a double contradic- 
since space is itself incapable of motion, much more a zero 
lagnitude. 
A geometrical line is defined as the path of a point, moving ac- 
cording to certain conditions, w^hich always limit its motion, in 



SYMMETRY IN SPACE. 3 

each of its positions, to one of two opposite directions. Or, it 
may be defined as a continuous series of all the points which fulfill 
certain conditions, among which must be the condition that each 
point is contiguous only to two others, one on the opposite side to 
the other. So also a surface may be deaned as the space in which 
a point moves, when, in each position which it assumes, a straight 
line may be drawn through it, and its motion be permitted, in any 
direction at right angles to that line, and in no other. Or, the 
surface may be defined as a series of points, through any one cf 
which a straight line may be drawn, such that all the contiguous 
points lie in a direction at right angles to that line. To either of 
these definitions of a surface, we must add, in order to make a 
geometrical surface, some other conditions which the points must 
fulfill. 

When the geometer has selected these conditions and would in- 
vestigate the form which the points, so conditioned, would enclose, 
he is not contented with the mere act of reason ; he endeavors to 
bring imagination to his aid ; to make a sensible image of the form. 
If he has been blind from his birth, he imagines his fingers feeling 
out the form ; otherwise he embodies it visibly, as in a drawing, 
or in a model. If he would convey a knowledge of it to others, 
he calls matter to his aid, and forces atoms of chalk, black lead, 
wood or thread, to fulfill approximately the conditions which his 
geometric law imposes upon the series of points. This dra^^ing, 
or model, is an expression of his idea, an enunciation of his law. 
A geometrical figure, whether upon the blackboard, or the printed 
page, or in a block of wood, or a set of stretched threads, is incon- 
trovertible evidence that a geometer has been expressing, by this 
means, a geometrical thought. 

The laws which please the geometer most highly are those 




4 GEOMETRY AND FAITH. 

'W'liich give us symmetrical figures, figures in which part answers 
to part ; either on opposite sides of one line or one surface, or 
about more than one line or surface. This taste is not peculiar to 
the geometer ; symmetry pleases the most savage, as it does the 
civilized man ; and men whose whole ability lies in other direc- 
tions, as well as the mathematician. A striking proof of the uni- 
versality of this taste was shown in the sudden and universal pop- 
ularity attained by the kaleidoscope. In a few years that toy of 
Brewster found its way to every parlor, and the heart of every 
child, ay, and every man in Christendom. Yet its sole magic 
consists in the symmetry which it imparts to a few fragments of 
irregular form. But that magic is sufficient to enchant all who 
come within its sway. We have never found any one uninterested 
in an extempore kaleidoscope, made by thromng open the piano, 
and placing brightly colored articles at one end of the folding lid. 

All regularity of form is as truly an expression of thought as a 
geometrical diagram can be. The particles of matter take the 
form in obedience to a force which is acting according to an intel- 
lectual law, imposing conditions on its exercise. It does not alter 
the reality of this ultimate dependence of symmetry upon thought, 
simply to introduce a chain of secondary causes, between the origi- 
nal thinking and the final expression of the thought. 

Many of the geometer's a 'priori laws were, indeed, first sug- 
gested by the forms of nature. Natural symmetry leads us to 
investigate, first, the mathematical law which it embodies ; then, 
the mechanical law which embodies it. Thus all the benefits which 
have come to our race from the pursuit and discovery and use of 
the keys of physical science, have been bestowed upon us through 
these suggestions of geometrical thoughts in the outward creation. 

But in the pursuit of mathematical knowledge men began, at an 



SYMMETRY IN SPACE. 5 

early age, to invent and investigate a priori laws, laws of which 
they had not received any suggestion from nature. And the 
intellectual origin of the forms of nature was made still more mani- 
fest w^hen these a priori laws, of man's invention, were, in many 
cases, afterwards discovered to have been truly embodied in the 
universe from the beginning; as, for example, Plato's conic sec- 
tions in the forms and orbits of the heavenly bodies, and Euclid's 
division in extreme and mean ratio. 

This division in the extreme and mean ratio was invented by 
the early geometers, without any known suggestion. It is evi- 
dent that this division might be illustrated in a great variety of 
ways. A whole must be divided into two parts, such that the 
first shall bear the same relation to the second that the second 
does to the w^hole. No matter what the w^hole is, a division of it 
approximately in this manner would be an expression of the idea 
of extreme and mean ratio. If the whole were a quantity (dis- 
tance, angle, surface, volume, value, time, velocity, &c.), and the 
relation were that of magnitude, the w^hole would be to the smaller 
part as unity is to half the difference between three and the square 
root of five. If, on the other hand, the whole were a work of art 
of any kind, or a system of thought, the relation would not be 
one of mere magnitude ; and the division would be a work of more 
ingenuity. But, whatever the whole, or the relation, the proper 
division would be an expression of the idea. 

Now we have, in nature, at least three embodiments of the law 
of extreme and mean ratio, two of which are very striking. The 
botanists find that two successive leaves, counting upward on the 
stem, stand at an angle with each other that is cither one-half, 
one-third, two-fifths, three-eigliths of the whole circle ; or some 
higher approximation to this peculiar proportion. The seed ves- 



6 GEOMETRY AND FAITH. 

sels and buds on a spike of broad-leaved plantain aflford one of the 
most instructive examples. They are usually set on a high ap- 
proximation, so that the order is not apparent. Take a piece of 
the spike, an inch or so in length, between your hands, and gently 
twisting reduces it to three ; w^hile a slight twist in the opposite 
direction brings out five rows, which a harder twist reduces to 
two. 

The efficient cause of this arrangement we do not know. It has 
been ingeniously suggested that it might be produced by a simple 
law of the genesis of cells. Let us suppose that each cell emits 
a new cell at regularly recurring intervals of time, and that the 
new cell begins to generate cells at the expiration of two intervals 
after its birth. A cell developing on a plane, under this law, 
would produce its cells in the phyllotactic order of the leaves, in 
the terminal rosette of a plant. But it is difficult to see how this 
hypothesis can be made to include and explain the whole phe- 
nomena of the arrangement. 

The final causes, although the devout mind always recognizes 
the impossibility of man's attaining a certainty concerning all the 
final causes of the phenomenon, are more obvious. It has been 
shown that this division of the circle insures in the only perfect 
way to each leaf its chance at zenith light, its best chance at air ; 
in short, that this phyllotactic law distributes the leaves most 
evenly about the stem. 

In the solar system, if we divide the periodic time of each 
planet by that of the planet next farthest from the sun, we shall 
have, beginning with the quotient of Uranus' year divided by 
that of Neptune and ending with the quotient of Mercury's year 
divided by that of Venus, a series of fractions agreeing very 
closely with the approximations of the phyllotactic law\ The 



SYMMETRY IN SPACE. 7 

problem was similar. The planets would not have remained in 
proper subjection to the sun had they been allowed to group 
themselves too frequently in one rebellious line, hanging upon 
the golden chain of his attraction, dragging him and themselves 
from their proper orbits. They must be kept evenly distributed 
about the sun ; and since they are moving, the times of their 
revolution, their angular velocities, must be divided by the same 
law as that which divides the stationary angles of the leaves. 

We have then in the plants a geometrical or angular illustra- 
tion, and in the planets an algebraical or temporal illustration, of 
the mathematical idea of extreme and mean ratio. The infer- 
ence seems irresistible, — these two illustrations, which cannot be 
imagined as having any causal or genetic connection, owe their 
intellectual relation to having sprung from One Mind. 

This is a striking illustration, but the same inference may be 
drawn from every form in nature, — planet, crystal, plant, and 
animals. All natural forms conform more or less closely to geo- 
metrical ideals ; sufficiently near to suggest these ideals to men 
fitted to receive the suggestion ; sufficiently near to show that the 
whole of nature may, in one sense, be regarded as a series of draw- 
ings and models, by which to teach the mathematics to students in 
the school of life. 

The final causes may never, however, be considered as wholly 
known. The perfection of the Divine workmanship is shown in 
the adaptation of each object in nature to a great variety of ends. 
The geometrical laws, on which the world is built, are adapted to 
all the wants and all the needs of every creature. Our liuniau 
needs are innumerably various, and nature finds means to satisfy 
them all. Our intellect craves symmetry, and through symmetry 
is first led to the perception of geometric law. But we love the 



8 GEOMETRY AND FAITH. 

symmetry before we perceive the law. The sense of beauty i? 
satisfied, even in externals, most perfectly, and fills us with most 
pleasure, in things that the understanding fails to analyze and 
define. Much has been written concerning an analysis of the 
beauty of outUne ; one great painter thinking it consists in flexure, 
others assigning it to a spiral, or a helix, or an ellipse ; while 
Darwin refers it to early association, while yet a suckling, with 
the form of the mother's breast. I venture with diffidence to 
give my own opinion, that the perception of beauty in outline is 
the unconscious perception of geometric law, — just as the per- 
ception of harmony has been demonstrated to be the un3onscious 
perception of arithmetical ratios in time, or algebraic law. The 
beauty of outline, I would say of external form, independently of 
expression, is in proportion to the simplicity of the geometric law, 
and to the variety of the outline which embodies it. Nor is it 
essential to the highest enjoyment of beauty that the conformity 
to geometric ideals should be perfect, any more than it is essential 
to the highest music to have the harmony perfect. On the con- 
trary, the higher degrees of beauty are apt to be found in forms 
that suggest, rather than embody, the ideal ; and especially in 
figures potentially, but not actually, symmetrical The monotony, 
which might result from unbroken regularity of form, is avoided, 
and a new grace is given, for example, to the higher animals, by 
their temporary disguise of symmetry, in their varied positions and 
movements. In the sea shells, the same end is attained by the 
spiral form, which so many of them take ; in which there is not an 
actual symmetry, but only a law of symmetry, the perfect develop- 
ment of which would require an infinite number of convolutions. 

In the forms of vegetative life, there is the widest departure 
from actual symmetry, and yet a constant suggestion of its laws. 



SYMMETRY IN SPACE. 9 

The phyllo tactic law secures to the tree a general regularity, and 
eq^ial growth upon every side ; and yet, by complication of detail, 
combined with occasional failure or destruction of buds, secures 
an endless variety of graceful forms, in each species. May we 
not then name beauty as another final cause, another end secured 
by the adoption of the division in extreme and mean ratio ? The 
approximations are beautiful to us, and the pleasure given to us 
was foreseen when the law was adopted. May it not also have 
been felt ; and may not the forms of flowers be but approximations 
toward the expression of an infinite beauty, hidden, from all finite 
sense, in the incommensurable ratio of that surd ? That the ex- 
ternal symmetry of animals may have beauty as its final cause, is 
rendered probable from the lack of symmetry in the viscera, which 
are hidden from sight. 

Whatever be our speculations upon such points, this at least is 
manifest, that the sense and the presence of beauty are kindly 
adapted to each other in the Avorld. Even shapeless matter de- 
clares its Creator's power; the perfect symmetry of crystalline 
forms, the potential symmetry of all the organic worlds, show forth 
His wisdom and His love. 



11. 



SYMMETRY IN TIME. 

Time has but one dimension, and is divisible only into before 
and after. In the zero of now, the future is becoming the past ; 
and this suggests the division of the future, and of the past, by the 
insertion of imagined presents, zero boundaries, dividing time into 
periods ; these imagined future nows becoming actual, as we suc- 
cessively reach them, and those past having been actual, as we 
passed them. As time flows only forward, the imagination runs 
backward into the past with the greatest difficulty ; indeed I am 
not certain whether it is possible for the imagination to run back ; 
when I attempt to do so, I find that I leap backward, by longer or 
shorter leaps, but never run continuously in imagination through 
time, except forward, from the moment to which I have leaped. 

By symmetry in time, therefore, we do not mean a similar ar- 
rangement of intervals before and after a certain moment. This 
has occasionally been attempted in 2^^^ recte et retro chanting ; but 
it is a transference of geometrical symmetry to time, where it is 
out of place, and tasteless. Symmetry in time is the arrange- 
ment of two or more similar series of intervals, to follow the same, 
or successive movements. When the set of similar intervals fol- 
lows the same moment, it constitutes keeping of time ; when suc- 
cessive moments, rhythm; unless the intervals are very short, 
when rhythm becomes tone, or color, and keeping time becomes 
harmony. 

(10) 



SYMMETRY IN TIME. 11 

The passion for harmony and rhythm is an essential element in 
human nature. It is a passion which varies greatly in intensity in 
difierent persons, but it is never wholly absent. Savage nations 
hav^e some rudiments of music and of song. The naked Fuegian, 
when the stormy winds of his inhospitable straits pause for a while 
in their wild uproar, chants his songs or hymns in a rude measure 
and melody. The dullest ear for harmony has an ear for rhythm 
sufficient to perceive the diflference betw^een prose and verse. It 
is when the intervals of time in rhythm become so short, as to be 
separately imperceptible, that the rhythm is called simply tone ; 
and harmony is the simultaneous movement of tones. Man finds 
pleasure in all forms of symmetry in time, whether the parts are 
perceptible, or imperceptibly short ; and the world has been made 
in exquisite adaptation to this taste of man. 

Space has three dimensions, time but one. Yet, in some re- 
spects, time is richer in its contents for man, than is space. The 
beauty of forms in space, is almost equalled by the beauty of 
color, and color arises simply from symmetry of times ; it is a kind 
of tone. Color, indeed, is more expressive, more directly produc- 
tive of pleasure to the eye, than form. The latter appeals more 
to the intellect, and is more directly expressive of intellectual 
ideas ; the former appeals more to the heart, and gives a sweeter 
pledge of the Divine Love. But beside color, symmetry in time 
gives us music in all its vast variety of forms and expressions. In 
music there is a beauty as distinctly intellectual as that of geo- 
metrical figures, and a power of expression which geometric form 
attains, scarcely even in the human figure and face. Nor can we 
omit to mention heat, which although not giving direct i-leasure 
to the mind, and the heart, as beauty, color, and music do, is 
still essential to the life of the body and to its comfort. Many 



12 GEOMETRY AND FAITH. 

chemical changes are also produced by minute symmetric mo- 
tions. 

A minute symmetrical division in space produces no sensation 
for us, except as it may lead to symmetrical motion. Thus the 
minute symmetry of the particles of a solid in a clear liquid solu- 
tion, is revealed to us only through the motion of light, and the 
changes which that motion experiences in going through the solu- 
tion. In like manner the temporal symmetry of motion in the ray 
of light coming from the stars, brings us more information than its 
mere property of making visible can give ; for, on cross-examina- 
tion by the spectroscope, it confesses the chemical secrets, and the 
degree of heat in the star at the time of its leaving, ages ago ; 
and also the direction of the star's motion. 

Man's organization, and his surroundings, are adapted to his 
love of music. His voice is capable of being regulated to musical 
tones of various pitch. The metals are sufficiently elastic to 
render them sonorous ; and, as the foundation of all, the air itself, 
by its elasticity, becomes the vehicle of sound and the instrument 
of music. 

Two gases, intermingled, remain to a certain extent indepen- 
dent of each other ; and, inasmuch as sound travels in each gas 
with a velocity proportioned to the density and elasticity of that 
gas, there will be, from a single source of sound, two sounds prop- 
agated in the mixture, with different velocities, interfering with 
each other, and destroying the pure tone of a musical sound. 
Now the atmosphere is a mixture of heavy oxygen, with lighter 
nitrogen. The elasticities are, however, so nearly adjusted to the 
densities, that sounds travel in either gas with nearly the same 
velocity, so that the air sounds in an organ pipe, as if one gas. 
Had sound travelled in these two gases at rates differing as much, 



SYMMETRY IN TIME. 13 

as the rate In them differs from that in most of the gases known to 
us, the use of wind instruments of music would have been impos- 
sible ; probably all music, even the tones of the human voice, 
would, in that case, have been discordant to an ear at any consid- 
erable distance from the source of sound. With the intense and 
elevating character of the pleasure derived, first from the tones of 
human speech, from the melody of birds, and other natural music, 
and secondly from the art of music, in our minds, we cannot but 
be grateful for this adaptation of the mingled atmosphere to the 
needs of man, in his higher nature. 

All undulatory motion produces a symmetrical division of time. 
The beauty of color, like that of tone, arises from an imphcit per- 
ception of rhythm. The harmony of tints in the landscape, 
like that of the sounds in a strain of music, arises from the har- 
mony of times in which the vibrations of the mediums occur. The 
pleasure, in either case, arises from an implicit, or unconscious, 
perception of keeping time. Heat also has its colors, or tones, as 
is known to all who have noticed that the sun's heat passes freely 
through glass ; which is impervious to the heat of a fire. What 
other advantages to man may hereafter be discovered, in this 
coloration of heat, time alone can show ; but when we consider to 
what an extent, through the providence of God, glass is employed, 
it seems not irreverent to own our gratitude to Him that this sub- 
stance reflects back the warmth of our apartments, and keeps it 
within, but allows the heat of the sun to pass through from with- 
out. 

Besides these hidden proofs of creative foresight, and benefi- 
cence, in the concealed, or minute symmetry of time, we sliall 
find open and abundant proofs in manifest rhythmic movements. 
In the play of alternating muscles, we perceive an adaptation of the 



14 GEOMETRY AND FAITH. 

physical frame to the intellectual taste. In walking, for example, 
there are few persons who do not feel the increase of pleasure and 
of power gained by keeping step. A single drum-tap, regulating 
the tramp of a large body of men, has sometimes an effect almost 
equal to that of music. The rhythm of verse, and of music, de- 
lights many who are comparatively insensible to both melody and 
harmony properly so called. Who that reflects upon the genius of 
Bach, of Handel, of Haydn, and Beethoven, and considers the 
eifect which music, such as theirs, has upon the world, can doubt 
the kindness of that superintending Power, who kindled the fire 
in their hearts, and through that in ours ; who also adapted the 
air, and the various materials, for man, by w^hich he pours out his 
musical conceptions ? Who that reflects upon the genius of a 
Homer, and a Shakespeare, and remembers to how many millions 
their verse has given delight and instruction, can doubt that the 
same Beneficence gave the poet his power, and men the heart to 
be touched with poetry ? 



III. 



NUMBER. 

Number is not, like space and time, matter and spirit, an inte- 
gral part of the Universe, nor is it a necessary attribute of either 
of these. Space and time are ^vithout parts or limits, and are, in 
themselves, so diverse that they would not suggest even the idea 
of duahty. Thus also amorphous matter suggests no number. 
Number is an impress of thought, it is a pure creation of Spirit ; 
and its constant suggestion in the forms and periods of nature, is 
a clear demonstration that nature is the work of an Intellect which 
controls both space and time in thought. The human intellect 
early learns number from the text-book of outward nature ; and 
dehghts in tracing, further than nature goes, the laws of number. 

From the great usefulness of this earliest abstract science, and 
from the fascination of its pursuit, arithmetic has, in modern 
schools, been allowed to usurp the place of geometry ; and the 
pupil has been taught to reason upon abstract numbers before he 
has learned to conceive clearly imaginary forms. From the same 
fascinating power, number has sometimes, in the minds of great 
men, like Pythagoras, been allowed to occupy a disproj.ortionate 
share of attention ; as though number included all proportion and 
beauty. Even the Hebrews, with all their clearer light of truth, 
appear to attach a mysterious power to number. 

There is a power in number. When our human thought at- 
tempts the survey of space and time, and would subdue these 

(IS) 



16 GEOMETRY AND FAITH. 

realms to obedience under our intellect, we find ourselves com- 
pelled, before we can attain any precision in our forms, to intro- 
duce number. The reason can deal, to some extent, with con- 
tinuous quantity, moving under continuous law, and not in the 
proportion of numbers. But the imagination cannot take a step 
with any clearness, much less can the hand build with any satis- 
faction, without referring quantities to a unit of quantity, to which 
the ratio shall be that of two numbers to each other. And of 
course our finite intellects handle with most ease the smaller num- 
bers ; so that these become to us the most important ; and there is 
not a number under ten which has not some strong associations 
with it in the human mind, which give it a kind of sanctity. 
These mystic charms cluster especially around the odd numbers 
three, five, seven, and nine ; which seem to have an individuality ; 
the first-named three being primes, while four is but two twos ; 
and six, two threes ; and these charms were felt in the earliest 
ages of human history. 

But nature also loves these numbers ; and they are illustrated, 
even to the untaught mind, by many phenomena ; organic beings 
possess a unity, which is absolute ; the sexes, of both plants and 
animals, give us duality ; the powers within, and those above, sug- 
gest the threefold division ; the points of the compass, the limbs of 
mammals, give us the number four ; the fingers of the hand, five, 
and so on. And the increasing knowledge of the physical world, 
in our nineteenth century, brings us increasing proof that God, 
who planned heaven and earth, was acquainted with numbers ; 
made all things in number, weight, and measure ; and adopted 
the smaller numbers, either out of preference for them, or in 
condescension to the minds of his children, whom he has placed 
here for their preparatory education. 



NUMBER. 17 

Chemistry is a science of this century, and it teaches us that, 
from the beginning, the numbers two and three have been domi- 
nant powers in the Universe. Simple unites with simple to form 
a couple, a compound. This couple rarely takes a third element 
to form a triplet. The couplets and triplets unite again in com- 
pound couplets, and thus the innumerable variety of substances 
is built up under the simplest possible combinations of number. 
Follow these substances through all their various modes of mo- 
tion and action ; in their weights, in their attractions, their gas- 
eous condition, their volume, their specific heat, their color at a 
high temperature ; and they are found still to be bound together 
by simple laws of arithmetical proportion. 

Consider also the law of extreme and mean ratio, as exhibited in 
the leaves of plants. In itself the law transcends the power of 
number, and had the plants fulfilled it with absolute accuracy, it 
might have been, even yet, hidden from the mathematician's eye. 
Bat the plants give it to us only by approximations ; approxima- 
tions which demonstrate that the exact law was known to the 
Builder of the plant, and is by him revealed to the mathemati- 
cian ; but which give to the unlearned the simpler conception of 
the first four prime numbers ; in the beautiful varieties of leaves 
opposite, and leaves ternate, five pointed and seven pointed stars. 

The laws of musical harmony are especially to be noted. AVhen 
the waves of the air are perceived only as continuous musical 
tones, and the individual vibrations arc not at all recognized, why 
should the ratio of four to five give us pleasure, and that o[^ eii^ht 
to eleven give us none? What process of education in our ances- 
try, what association of ideas, renders the effect of the one com- 
bination harmonious, of the other discordant? Any attempt to 
explain it will but strengthen the conclusion, that to the I'niKler 



18 GEOMETRY AND FAITH. 

of the ear the laws of number were known, and that the ear was 
constructed with reference to them. 

The harmonies of light and heat are not sufficiently well under- 
stood to make the argument here so apparent. Yet there is, 
doubtless, in these departments also, an adaptation of the human 
sense to the perception of effects arising from simple numerical 
proportions in the frequency of vibrations. In the matter of 
geometric form, while the value of proportion has been felt by all 
artists, and all architects, the value of numbers in the proportions 
has not been universally conceded, nor its place assigned. Yet 
I have by experiments, upon unprejudiced persons of good taste, 
strengthened greatly my inclination to accept Hay's law, — that 
angles, real or potential, are the essential elements of geometric 
beauty ; and are beautiful in proportion to the numerical sim- 
plicity of their ratio to the right angle. 

With these manifest indications that the divine thought, the 
ideals of the creation, include number as an essential element, we 
may well understand the enthusiasm of early thinkers over the 
properties of the smaller numbers. The sacredness of the num- 
ber three has been made especially prominent in Christendom. 
The four elements of the ancients, and Erigena's fourfold division 
of nature, show the power of the points of the compass to impress 
their number on the human mind. The five digits of the hand, 
and the prevalence of fivefold divisions in the floral kingdom, give 
us the five-pointed star with its symbolism ; point up, for manhood 
and virtue; point down, for beastliness and sin. The lily tribe 
gives us the six-pointed star ; and six, a perfect number, in which 
the sum of the factors equals the product, is fitting as a symbol of 
the descent of the divine into the human trinity, the indwelling of 
God in man ; the Perfect perfecting his child. The seven notes 



NUMBER. 19 

of the diatonic scale, the seven distinct colors, and other natural 
examples, fall in with the seven days of the week, the quartering 
of the moon's period. Jew and Gentile alike have hallowed the 
number seven, and no other number occurs so frequently with 
sacred associations in Jewish and Christian literature. Higher 
primes than seven do not enter much into our human thought, nor 
appear to be embodied distinctly in any part of creation known to 
us. The weeks in the year are four times thirteen ; that is, there 
are about thirteen moons in the year ; the only example I re- 
member of a prime number above seven prominently suggested 
by nature. The nine muses, the ten numerals, the twelve months, 
and twelve apostles are numbers not prime. 

Music, painting, the coloring of nature and art ; architecture, 
sculpture, drawing, the beauty of proportion and form ; how large 
a portion of our earthly pleasure and spiritual culture depends 
on these ; and these draw their charm in some mysterious way 
from the numbers two, three, five, seven. The number of prime 
numbers is unlimited ; and since the first four give us, in the har- 
mony of tones and colors, and in the proportions of form, such 
varied sources of high pleasure, such potent modes of spiritual ex- 
pression, Ave may reverently hope, that in the immortal life, the 
same Beneficent Power which makes two, three, five, and seven, 
thus minister to our joys below, will open to us more of the infinite 
treasures which lie hidden in the boundless fields beyond. 



IV. 

THE CALCULUS. 

Space and time are so entirely diverse in their nature, that 
there is no connection or relation between them ; except through 
the mind, as percipient of both ; or through will, manifesting itself 
in motion. Li contemplating space we see it as external to the 
mind ; our consciousness does not sharply locate its own where- 
abouts ; we fancy ourselves near the Eyegate or Eargate of the 
town of Mansoul; but cannot say precisely where our council 
chamber may be situated. Not so with time, our consciousness is 
sharply defined ; we are neither in the past nor in the future ; our 
conscious moment is the now, without duration. Hence we can 
more readily imagine ourselves freed from limitations of space than 
from those of time. We can imagine to ourselves time in the flow 
of our own thoughts ; the thought of space necessarily takes us 
out of ourselves. But when we go out of ourselves and contem- 
plate space, we carry time with us in the very action of our 
thought. In all closer contemplation of outlines, the attention is 
transferred successively to different points of the figure, and time 
is occupied by that transfer. Thus w^e come naturally, and almost 
inevitably, to regard the line as the path of a moving point, the 
surface as generated by a moving line. 

Thus space and time, though heterogeneous, are united into one 
science of mathematics by human thought ; and the laws of alge- 
bra, or time, are applied to geometry, or space. By this simple 

(20) 



THE CALCULUS. 21 

device, into which Descartes and Newton were led by nature's own 
guidance, the human mind has extended almost indefinitely its 
geometrical acquisitions ; it was by carrying, as it were, its native 
element of time with it into the domain of space that it lias con- 
quered so vast a field. 

When we remember how intense the delight which man feels in 
the discovery of mathematical truths ; how many of the noblest 
thinkers of the race have owed their finest discipline to this pur- 
suit ; how rich the harvest of practical benefits which have flowed 
from the application of mathematics to the arts and sciences ; how 
magical their efiect has been in banishing superstition, and elevat- 
ing the general tone of human thought and human endeavor, we 
may surely own, with gratitude, the marks of divine wisdom and 
love, in this gift to man, of the power to penetrate space, and a]> 
ply to it the laws of time. It is a peculiar gift, not a necessary 
accompaniment of intellect, for sometimes the brightest intellects 
possess it in only a very feeble degree. Thankfully, therefore, do 
we acknowledge the presence of an Infinite Spirit, giving good 
gifts to man in the inspiration of a Leibnitz and a Lagrange, as 
well as of a Handel and a Shakespeare. 

The main source of this power given by algebra to the geometer, 
is the comprehensiveness of the language put into his hands. The 
introduction of general and abstract terras is always a means of 
enlarging the grasj) of thought, and increasing the clearness of 
reasoning. Space has its three dimensions, its elements of magni- 
tude and direction ; and although, in one aspect, the simplest of 
all possible objects of thought, may yet, for purposes of reasoning 
concerning it, be advantageously reduced, by algebraical langunge, 
to the one term of quantity, capable only of flowing in one direc- 
tion, and being considered as greater or less than a 



22 GEOMETRY AND FAITH. 

tude. But the generality thus introduced is made vastly more 
general by using symbols which shall combine, in one letter, vari- 
ous forms and relations in space, defined according to judiciously 
selected and easily interpreted laws. Thus, for example, all possi- 
ble triangles, plane and spherical, and all their properties are im- 
plied in the single equation, r = pq ; and a similar condensation 
of meaning is attained in mechanical science. Another source of 
the peculiar power of the calculus arises from the plasticity which 
it gives to infinitely rigid space. In experimenting upon a rec- 
tangular beam, cut from a round piece of timber, we can readily 
determine its strength when set edgewise ; but cannot tell what 
the strength would have been had the sides been in different pro- 
portions. The rectangular parallelepiped inscribed in a cylinder 
is as absolutely fixed in its dimensions as the hewn timber, but by 
expressing those dimensions in language borrowed from the science 
of time, we can imagine them changing in their proportions, and 
the strenoith chan<2:in<2: with them. Thus we can determine the 
precise proportion they must bear in order to give the strongest 
possible rectangular beam that could be cut from a round log. 
This illustrates, by a simple example, the power given to geome- 
try by Newton's conception of fluxions, his introduction of the 
idea of velocity into the consideration of form. 

The appearance of the same algebraic law in the creation, under 
the two forms ^f time and space, has already been alluded to as 
proof of unity of design ; the angles of leaves and the angular ve- 
locity of planets being expressed by the same series of fractions. 
Other examples confirm the subhme induction. The elasticities of 
gases, strings, and rods are so fundamentally diflFerent in kind that 
we see no connection between them. The elastic force of the 
stretched string we need not determine ; that of the rod, and that 



THE CALCULUS. 23 

of the gas, can be determined only by experiment, and when deter- 
mined they have no very apparent connection or relation with each 
other. Nevertheless, each of the three has a peculiar relation to 
the force of gravity ; of which it is, nevertheless, entirely indepen- 
dent. The velocity of a sound traveling in the air, near the earth, 
w^ould be, were no heat developed in the action, equal to the velocity 
acquired by a body falling from a height equal to that which the 
atmosphere would have could it be all compressed to the density 
of that near the earth's surface. The velocity of a wave traveling 
on a string is equal to that which would be acquired by a body fall- 
ing from a height measured by the length of the same, cord equal 
in weight to the tension of the string. And if we take a very 
fine glass thread by its two ends, the infinitely varied and beauti- 
ful forms which it can be made to assume, of waves and folds and 
kinks and loops, the figure eight and the circle, are all expressed 
in mathematical language by the same forms as those which ex- 
press the motions of an ordinary pendulum, under the forces of 
gravity. The genetic connection, between these forms and these 
motions, we do not see, any more than that between the times of 
the planets and the angles of the leaves, but the intellectual con- 
nection we detect, and it leads us to recognize with reverential 
awe the presence of Intellect in the disposition of the particles 
of both gaseous and solid bodies. 



V. 

APPLIED MATHEMATICS. 

Pemect symmetry belongs only to the ideal, not to the actual. 
The algebraic conditions are exactly fulfilled by points of space, 
in an invisible and eternal reality ; to this real form, conforming 
to the algebraic ideal, the material embodiment makes at least a 
rude approximation. The algebraist devises conditions of various 
degrees of complexity, delighting chiefly in the simplest ; and 
especially in those giving, with simplicity of conditions, the 
greatest variety of resulting forms. 

The symmetrical forms of nature suggest to man the invention 
of laws of symmetry, at first simply to explain nature, then to an- 
ticipate her work ; leading to new examinations of that work. 
Thus the great mathematical sciences have been alternately the 
creation and the creators of physical science. The physicists 
have been prone to deny that the mathematics constitute a sci- 
ence ; they have inclined to pronounce them only a key to science, 
a convenient language wherein to discuss the problems of matter 
and motion. The mathematicians, on the other hand, whom we 
should naturally consider the best judges of what their own work 
is, have declared that geometry is the science of space, algebra 
the science of time, and that these are simply the first subjects 
handled by the human intellect with sufficient freedom, vigor, and 
precision to enable us to draw necessary conclusions. As for 
geometry and algebra being mere keys to physics, the mathema- 



APPLIED MATHEMATICS. 25 

tician would sooner declare the whole visible creation a mere set 
of models and diagrams wherewith to illustrate the laws of space 
and time. Whichever of these conflicting views is right, it is un- 
questionable that the highway to the temple of truth leads alter- 
nately from mathematics to physics. Observation alone can lead 
to nothing, without insight, — without that clearness of inward 
vision which sees more than the outward fact, sees the divine ideal 
which the fact partially embodies. 

Now in this sublime ascent to knowledge the first steps are 
easiest, and the way to them has been made exceedingly plain 
and attractive. '' In the beginning the Creating Spirit embodied, 
in the material universe, those laws and forms of motion which 
were best adapted to the instruction and development of the 
created intellect." The circle and the ellipse are among the sim- 
plest of figures, defined by the simplest laws. Accordingly the 
Creator has strewn examples of the circular form around us on 
every side ; and, by the pictured alphabet of the heavens, called 
our attention to the consideration of elliptical orbits. When, in the 
course of ages, some of the comparatively easy problems of as- 
tronomy had been successfully solved, problems of more difficulty 
were gradually brought into view ; and phenomena which were 
not obvious, not pictured aljjhabet, but the fine ])rint of creation, 
led men into the hidden knowledge of optics, electricity, chera 
istry, and other forms of molecular physics. The course of 
history and of scientific progress has been precisely what it might 
have been had God designed to educate men ; to reason with 
them and teach them the sciences ; for there has been a con- 
stant presentation of simpler truths, whereby men have been 
led to the acknowledgment of those less obvious : and this is 
essentially reasoning. 



26 GEOMETRY AXD FAITH. 

Four centuries before the Christian era, the mathematicians of 
Greece were lured into the study of the conic sections ; and this 
prepared the way for the mathematicians of later ages to discuss 
fully certain equations of the second degree. These were suf- 
ficient for all the more obvious phenomena of astronomy and me- 
chanics ; and as the demand for higher mathematics has been 
made by physics, the supply has been granted. The faith, which 
prompts the scientific investigator to his labor, he may never have 
expressed in words, but his actions show us what it is, — an in- 
born, ineradicable conviction that the outward universe is intelUgi- 
ble, and shall at some day be understood. But that day ever re- 
cedes, into the glorious future, as we approach it ; the rate of 
scientific progress increases from decade to decade, and yet the 
new problems, and the new instruments for their solution, in- 
crease more rapidly. The Divine Intellect can never be ex- 
hausted by the human. 

A more detailed examination of the history of the separate 
sciences would only confirm our conclusion, that, in the selection 
of laws under which to subject the universe, God has chosen, for 
those things which would first press themselves upon man's atten- 
tion, those which are most readily interpreted by man's intellect ; 
and employed more intricate laws for things which would natur- 
ally escape man's notice until the state of mathematical science 
enabled him to take higher problems ; in which we recognize evi- 
dence of that kindness and foresight, that care for our education 
and our growth in knowledge and wisdom, which is an inspiring 
pledge to us that we are indeed children of the Most High. 



VL 



MOTION. 

The universe about us is in motion. Nothing on which the eye 
can fall, or the existence of which the hand of science can demon- 
strate, is at rest. The sun rises and sets, the moon waxes and 
wanes, the very stars are in motion, to the telescopic eye. Clouds 
drive over the heavens, and billows roll over the deep ; vapors rise 
from the ocean, rivers run to the sea, and the free winds play 
around the globe. Plants are ever growing or decaying ; and 
animals maintain their waste, or their waste consumes them. Our 
modern theories show that the sensible properties in inanimate and 
apparently motionless matter, such as temperature, color, weight, 
are really modes of motion in the particles of matter ; and this 
re-echoes the sublime statement of the earliest seer, that the intro- 
duction of motion into the universe was the first act of creation. 

For, upon a closer examination of motion, and more accurate 
investigation of its laAvs, what do we find ? That the first law of 
motion is this : A body, free from external influence, moves with 
uniform velocity in a straight line forever. This is the first law of 
motion, derived from the widest generaHzations, by legitimate in- 
duction from observations, on an immense variety of motions, in 
nature, and in the laboratory. But to what an astonishing result 
does this law lead us when we apply it to the case of a body at 
rest, the velocity of which is nothing. A body at rest, free from 
external influence, would remain at rest forever. In other words, 

(27) 



28 GEOMETRY AND FAITH. 

the first result from the scientific observation of motion in matter 
is, that matter cannot move. Hence follows the inevitable con- 
clusion, that the cause of all the motion in the universe, is some- 
thing else than matter. Higher than this the mvestigation of 
motion itself cannot lead us ; but this is high enough for a most 
valuable stepping-stone. 

Why do we ask the cause of motion ? Whence do we derive 
the idea that there is a cause for it ? It is not simply the impos- 
sibility of our imagining a beginning ; the beginning of motion we 
often see. But the motions which we most narrowly examine are 
those produced by our own w^ill ; we are conscious that our own 
volition is the cause of such motions ; and this consciousness is the 
foundation of our faith that motion always has a cause. Is this 
foundation trustworthy ? Beyond all question it is. Nay, it is 
the foundation of all possible physical science ; no man can ex- 
tend a generalization beyond the particular instances for which he 
drew it, unless he leans on this consciousness of causing. To 
return to motion, — matter cannot move, our will can move it ; 
there is nothing to suggest any other origin for motion than voli- 
tion ; hence we naturally, and legitimately, infer that the motion 
which we see, everywhere in the universe, is produced by a will, 
independent of matter, and superior to all the phenomena. 

Thus the first law of motion, established in the earliest revival 
of science, demonstrates not only the existence of God, but his 
perpetual presence and action. Every moving thing in the 
heavens, or on the earth, bears the same sort of testimony to his 
being and presence, as that borne by the human voice and action 
to the presence of a man. Whenever we see anything in motion, 
God is the mover. In the ancient tongues this was one of his 
names. The winds blow at his command, the sun rises because it 



MOTION. 29 

is his will, the falling rain and running stream are his gift ; and 
each beating pulse, each breath that we unconsciously draw, is 
a proof that this machine of the body is, each moment, depend- 
ent on the sustaining love and power of its Creator. 

Since we thus refer all motion, even that in our own frames, to 
the will of God, it may be thought that we are destroying man's 
freedom, — making him a mere machine, kept in motion by the 
Maker's supervision. But this objection to the doctrine of man's 
present dependence, forgets that the consciousness of our freedom 
is the very basis on which we have built our faith in the existence 
of God. It is from our own consciousness of power, to cause mo- 
tion at our own will, that when the first law of motion has ex- 
cluded us from ascribing it to powers inherent in matter itself, we 
ascribe all motion to his will, rather than to any unconscious 
natures. 

This consciousness of our own power, our own will, may be de- 
nied in words ; but it will presently betray itself, lurking in the 
mind ; it cannot be really denied ; it is the foundation of all phi- 
losophy and faith. The body, in all its molecular changes, by 
which ultimately the free movement of the limbs is produced, is 
moved by the will and power of God ; the first law of motion 
proves that ; yet the direction of the movement in the limbs is 
with man, consciousness testifies directly to this ; man is free, and 
cannot heartily believe himself to be otherwise. 

Our muscular power is not ours, but it is, to a certain extent, 
under our control. We caimot lift a finger without the aid of liini 
who formed us ; and yet it is we who move our hands. So the 
engineman, who has not, in his own muscles, strength to drive a 
single loom, yet, by controlling the valves of his engine, keeps the 
machinery of many spindles and looms in motion. Thus, with all 



30 GEOMETRY AND FAITH. 

man's frailty, and his absolute dependence upon other powers, he 
yet remains a cause, — free and efficient to control and direct the 
engine of his body, wonderfully framed and intrusted to his care. 

Our argument has been, thus far, drawn only from the uniform 
velocity of motion ; but the second clause in the first law would 
lead to the same result. A moving body, free from external in- 
fluence, moves, not only with uniform velocity, but in a straight 
line forever. 

As we have no apparent examples in nature of a uniform veloc- 
ity, so we have none of uniform direction. External influences 
perpetually accelerate or retain the velocity and change the direc- 
tion of moving bodies. But as the first part of the law is derived, 
not from actual examples or instances, yet by irresistible induction 
from observed facts, so the second part follows by like unavoida- 
ble inference from phenomena ; indeed, both parts are defended, 
by some mechanicians, as axioms, needing no other proof than 
Leibnitz's principle of the sufficient reason. 

As the varied motions of the universe cannot have sprung from 
the action of matter, that being inert, so the constant changes 
of direction in the motions prove that the forces, indeperdent of 
matter, are still acting. The rebounding of a solid from a solid 
shows that the particles of the solid adhere by some form of force 
diflerent from a cohesion of contact, — elasticity implies that the 
particles are held together by some force which permits their dis- 
tances from each other to vary within certain limits. When the ball 
leaves the muzzle of the gun its path instantly begins to be con- 
cave toward the earth ; and would be so at any conceivable degree 
of velocity. The meteor passing the earth at eighty miles a 
second bows to her as he passes. Thus the moon also is perpet- 
ually deflected from its path by the earth, and the earth by the 



MOTION. 31 

.moon, and both are turned constantly aside from their straight 
course by the sun ; and the whole host of heaven is constantly 
moving in a rhythmic dance, wherein each star influences the 
motions of the whole, and is influenced by the movements of 
each of the others. 

Our consciousness that w^e cause motion leads us to ascribe all 
change of velocity to force, all fc/rce to will. The same conscious- 
ness bears witness also that all change of direction implies the 
influence of will. The w^eight of bodies, the attraction of gravita- 
tion, the correlated forces of the universe, these are but reverent 
forms of words in which we speak of that which can only be re- 
ferred to the Divine Will. The untaught man, the poets of the 
earlier ages, were more true to reality when they used more re- 
ligious forms of speech. It is not so much figurative, as literally 
true, to say that He who formed the Seven Stars and Orion 
still guides them on their w^ay. Their circling orbits by their 
figure, and the golden orbs themselves by their motion, continu- 
ally manifest the presence of His guiding hand. The forces 
of cohesion and repulsion, of electrical and chemical change, of 
heat, of light, — all of the forces by which the existence of a 
particle of matter can possibly make itself known to our human 
senses, are but manifestations of the living action of the Most 
High. 

Thus the first law of motion leads us to see God in all things, 
and all things as the present creations of his hand. It might 
lead us astray, it might lead us to Pantheism, were it not that it 
first leads us to perceive that force is an attribute of will, and in- 
dependent of matter ; thus keeping us to the conclusion that tlie 
Creator and Governor of all things is free, living, — and our 
hearts add, good. 



VII. 



MUSCULAR ACTION. 

We \ ave spoken of the human frame as an engine of wonder- 
ful construction, whase movements are made dependent on the 
human will. Yet it is manifest that more of its motions are in- 
dependent of the will than are dependent upon it. The involun- 
tary muscles, and the involuntary movements not muscular, are 
those which are essential to the very existence of the body. The 
circulation of the blood, and its purification through the alternat- 
ing expansion and contraction of the chest, are obvious instances 
of these vital, involuntary actions. Not less important is each one 
of a thousand hidden operations, — capillary movements, glandular 
secretions, the removal of the^effete and the replacing of the liv- 
ing molecules; to say nothing of more muscular actions, — the 
peristaltic motions, and the wonderful unconscious artifices of 
swallowing, coughing, sneezing, and the like. 

The voluntary muscles are also capable of involuntary action. 
This is shown not only by occasional convulsive twitchings, or 
more violent convulsions, but, in a still more instructive manner, 
by inveterate habit, operating in sleep, or even when the will op- 
poses. But, although the action which has become habitual is not 
done from distinct, conscious volition, the habit is originally formed 
by acts in obedience to the will. The law by which the voluntary 
action becomes involuntary habit, although it reduces, too fre- 
quently, man to f^lavery, is truly beneficent in its design, and in 

(32) 



MUSCULAR ACTION. 33 

its best effects. The simplest statement of the law Avould, perhaps, 
be found in saying that actions which have been associated in vo- 
htion become associated in execution. In other words, when we 
have done several things at the same time, or in quick succession, 
the attempt to repeat one of these actions will tend to produce an 
involuntary repetition of the others. For an illustration of tlie 
beneficent action of this law, we may take the skillful player upon 
a musical instrument, who is conscious of a volition only at the 
commencement of each musical phrase : the fingering of the sep- 
arate notes comes from associated execution. In a familiar piece 
his volitions would be even less frequent, being necessary only at 
the commencement of a new strain. 

This case of the skillful player, being less frequent, seems the 
more striking ; but there is scarcely an action in life which is not 
aided by the beneficent operation of the same law, just as there is 
scarcely a mental action which does not illustrate the kindred law 
of the association of ideas. The child, just learning to walk, 
makes a painful effort at each successive movement of each mus- 
cle called into action. It requires all the concentrated energy of 
his will to make the successive volitions necessary for simple stejv 
ping from chair to chair. But in a few months he is able, by as- 
sociated execution, to set in action, by a single volition, a series of 
alternate motions, that carry him forward, without his attention, in 
a given course, at a uniform speed. No power of will is re({uircd 
in walking, except when we wish to alter the velocity, or the direc- 
tion, of our movement. 

When the successive movements, dependent on associated ex- 
ecution, are connected, as in walking, by a law of simple alterna- 
tion, the case is not difficult of explanation ; and the physiologists 
show us how the will relieves itself from duty by a switch, turnin;: 



34 GEOMETKY AND FAITH. 

off the currents of sensation and command from entering the main 
office in the brain. 

Other cases, in which the operation of the law is no less impor- 
tant to our comfort and convenience, require, however, much more 
intricate combinations of movement. In many familiar occupa- 
tions we require our hands to guide an instrument rapidly and 
freely through outlines of complex, but definite, forms ; as, for ex- 
ample, in writing, or in free-hand drawing. All men have more or 
less of this power to execute ideal figures, or to imitate given 
forms. This power has been gained, like that of walking, only 
through repeated and laborious efforts. When moving the hand 
in one direction, we need a new volition to change its direction, 
or to alter its velocity ; hence our first attempts at curvilinear mo- 
tion produce polygonal lines. In order to produce a curve, as in 
the ordinary forms of the capital letters, we must produce, by sev- 
eral muscles acting at once, motions in several directions at the 
same time, each movement varying in velocity according to defi- 
nite laws. To draw, for example, a circle, by any conceivable set 
of muscles operating on the arm, would require at least three sets 
of muscles, each acting in a different direction, and no two ex- 
actly opposed ; and these would be obliged to accelerate and re- 
tard their action by peculiar laws. The circle, however, is the 
simplest of all curves : in the ordinary operations of writing and 
drawing, the rates of acceleration and retardation must follow 
more comphcated laws. Of these laws we think nothing, we know 
nothing : we see the curve which we would form, and a single im- 
pulse of the will sends the pencil along the waving oathne. 

Charles Babbage, a successor of Sir Isaac Newton in the 
Lucasian chair, has won an immortality of fame by inventing a 
machin 3 which will tabulate in numbers the results of any alge- 



MUSCULAR ACTION. 35 

braic law which it may be set to obey. But how much mjre won- 
derful is this calculating engine of the human body, which is not 
confined to arithmetical results, nor does it require that its director 
should be learned in algebraical notation to set it at its appointed 
task, but Avhich is set by the artist, with his delicate perception of 
the bea'jty of form, to embody his divine ideal ! and it obeys, and 
places before us on the canvas, those figures, which, unconsciously 
fulfilling algebraic or numerical law, reach far higher, and express 
the spiritual thoughts and purposes of the Master. Is not the 
Maker of this wondrous engine of the human body worthy of grat- 
itude and adoration V 

Consider how wonderful is the phenomenon of a boy's throwing, 
successfully, at a mark. The epicycloidal theories of Ilipparchus, 
the Newtonian theory of gravitation, the resolution of centripetal 
and centrifugal forces, the conic sections of Apollonius, the modi- 
fications of those curves by the resistance of the air, — all these 
are involved in the problem, and must be practically solved, with 
considerable accuracy, before the school-boy can give his fellow a 
good ball, or catch one on the fly. 

It may be observed that the mechanical contrivance by which 
the human hand is enabled to go through all imaginable motions, 
and strike, at a free sweep, any curve, however complicated or 
however beautiful, is an embodiment of one of the most celebrated 
of mathematical conceptions, discussed in the writings of Plato 
and Aristotle, constituting, in its development, one of the chief 
triumphs of Ilipparchus, and brought by modern matliematicians, 
through the arithmetic of sines, and the canon mirijicuB of Nai)ier, 
into a form capable of reducing to a regular curve the most vaii- 
able and irregular table of observations. In this method of epi- 
cycles, as used by the modern computer, a series of arms is sup- 



36 GEOMETRY AND FAITH. 

posed to be carried, each on the extremity of the preceding, and, 
during the revolution of the first, each to revolve once oftener than 
the preceding ; that is, while the first arm of the series revolves 
once, the second revolves twice, the third three times, and so on. 
It only remains for the computer to fix the length of these arms, 
and determine their original position, in order to make the end of 
the third or fourth, or, in cases of difficulty, of the fifth and sixth, 
describe any path he wishes. In the human limb, the upper arm 
is the first, the fore-arm the second, the hand the third, and the 
fingers the fourth, fifth, and sixth of these rotating arms ; and 
the fixedness in the ratio of their length is more than compensated 
for, by our ability to graduate the ratio of their revolution at will. 
Is there no meaning in the fact that the most cunning device of 
human ingenuity for making a point travel, under simple laws, 
through the greatest variety of paths, should thus prove to be sub- 
stantially the same with that adopted in the very creation of the 
human frame, for enabhng the hand to guide its tools with freedom 
and accuracy ? 



VIII. 

GEOMETRICAL INSTINCTS. 

Since our fellow citizens of all the animal kingdom are, like 
ourselves, dwellers in space and time, it is necessary for them, 
also, to have ideas of distance and direction in space, duration 
and lapse in time. Ideas gained by sense-perception seem to fur- 
nish them, as us, the data for reasoning ; but ideas of direct intui- 
tion do not appear to afford to them, as to us, objects Avhereon to 
reason ; but merely serve, as certain of the kind do for us, as the 
stimulus of desire, and the guide of unreflective action. These 
intuitive ideas, perceived by inward sense, but not, perhaps, dis- 
tinctly eliminated in consciousness from co-existent ideas, are, in 
the lower animals, called instincts ; and when used in like manner 
by us, not as propositions for conscious reasoning, but as the 
grounds of instantaneous judgment or action, they have the vari- 
ous names of instincts, feelings, promptings, conscience, or genius, 
according to the nature of the objects to which they relate. 

Geometrical instincts are common to us with all the animate 
races. That instinctive trigonometry, for instance, by which a 
child, of a few months old, learns to tell the position of any object, 
to which his two eyes are directed, is probably exercised by all 
animals with two eyes capable of being turned upon a single object. 
The most striking instance is popularly believed to be the young 
^uail, which is said to run, as soon as hatched, freely about, peck- 
ing at mirmte oTyects with as true an aim as its mother's. I have 

(37) 



38 GEOMETRY AXD EAITH. 

seen a Setter's pup, sired by a Pointer, when a few weeks*' old, 
point at a piece of anthracite with all the accuracy of its father, 
which it had never seen. In 1843, a toad, frequenting the gar- 
den at Burgovne's Headquarters, in Cambridge, and losing by an 
accident the sight of one eye, was for a long time unable to aim 
his tongue, with certainty, at the overloaded bees, who, returning, 
missed the threshold of the hive, under which the toad, expecting 
such misfortunes of his insect neighbors, was accustomed to sit and 
await their fall. In time he, like human beings who have had like 
accidents befall them . learned to substitute optics for trigonometry, 
and instead of solving triangles, with a base and adjacent angles 
given, decided on the position of objects requiring a certain focal 
adjustment and direction of his remaining telescope. 

But how comphcated the action by which he proved this, if I 
may use the expression, unconscious knowledge of the position of 
the bee 1 A single conscious volition, and his tongue, which is 
rooted in the front of his mouth, with the tip lying far down within 
his throat, flies out and back like an electric spark, having taken 
the bee up on its tip, and thrust her down the throat of the toad. 
His calculating engine, set by the adjustment of his eyes, not only 
computes the exact curve in which the tip of his tongue must 
move, but the exact force and velocity with which it must be 
sent, in order to accomplish its mechanical errand. The same 
marvelous unconscious calculation is proved when a boy hits the 
mark, at which he aims a stone, or when the expert player at bill- 
iards strikes his ball on exactly the right part of the ball, in 
exactly the right direction, and with exactly the right force, in 
order to make it pursue a long course, partly curved and partly 
straight, with rebounds from the cushion, and rebounds from other 
balls, and come to rest at a determined place. He does not know 



GEOMETRICAL INSTINCTS. 39 

the difficulties of the problem he has solved ; he does it with as 
little of conscious calculation as that with which the toad snaps 
up the bee ; but this only renders the more striking the wonderful 
perfection of the muscular and nervous organization, as machinery 
adapted to describe geometrical figures anc solve mechanical prob- 
lems of great perplexity. 

The architectural or nest-building instincts of animals show the 
geometrical and mechanical knowledge of the Creator of animals 
in a very conspicuous manner. Men invented and used the arch 
long before human mathematicians solved its theory. Many other 
of our mechanical inventions, and some of them, as the barrel and 
the potter's wheel, for example, of a wonderful kind, have an 
antiquity that long antedates abstract mathematical thought. We, 
reasoning, discover the principles underlying our inventions, and 
thus improve science, which again suggests new inventions ; so 
that human art and human science stimulate and foster each other 
to endless competition and endless progress. The lower races have, 
apparently, no abstract thoughts, no intuitions, that arc brought, 
with consciousness, among the data of their reasoning ; in other 
words, they appear to have no science, and hence their progress 
in the arts is so slow as to appear stationary. But their instinc- 
tive judgments appear, frequently, more accurate and wonderf il 
than those of men. To see the republican swallow, coming through 
the air, fold her wings at precisely the right moment, and when at 
precisely the right speed, in order to enter softly and smoothly 
her earthen bottle, makes the art of the most skillful coxswain 
seem rude in comparison. The weaving of the bird's nest is in the 
case of the African grosbeak carried to a degree of perfection that 
vies with that of the nicest works of man, unaided by machinery. 

But the architectural work of insects is most wonderful, and 



40 GEOMETRY AXD FAITH. 

none more so than the familiar honeycomb. Always admitted by 
men. from the earhest ages, it was, at the beginning of the last 
century, discovered by Maraldi, of Nice, to embody distinctly the 
complicated geometrical conception, of forming cells to contain a 
fluid mass, with the greatest strength, the greatest economy of 
space, and the greatest economy of material. The paper-making 
wasps make rude approximations toward the solution of the same 
problem, but inasmuch as the larger part of their material is 
cheaply gathered from the surface of wood, there is no call for so 
strict an economy. The bee. needing a water- proof material, yet 
finding the resin of trees too adhesive to be worked with facihty, 
confines her use of such resin to the places in which she needs 
especial strength, or especial resistance to moisture ; and, for her 
ordinary work of cell building, uses a material wholly secreted by 
the glands of her own body. Her cells are approximately hexa- 
gons, which hold more, and have shorter, and therefore stronger, 
sides than any other figures which could be packed without waste 
of room. They are set, with economy of room, base to base ; and 
still further streno;thened on the bases, bv beinoi; set one ao;ainst a 
part of three, so that the bottom of each cell is supported by three 
partition walls on the other side. Finally, and most curious, the 
bottom of each cell is depressed in the centre to about that degree 
which will save most by diminishing the height of these supporting 
partitions without increasing too much the area of the floor which 
rests upon them. I say about that degree ; and the accordance 
of the average cells, in a normal piece of comb, with theoretical 
perfection as determined by the calculus of Newton, is very close. 
AVe should not expect perfection, because the perfection of the 
artisan is to be measured, not by the perfection of his results, but 
by the perfection of their adaptation to his end. He were a poor 



GEOMETRICAL INSTINCTS. 41 

farrier who polished his shoes with the care that a dentist bestows 
upon his gold filUng. Nor would the bee be a wise economist if 
she wasted time in bringing to theoretical perfection her saving of 
wax. What the bee's conscious aim is, in the construction of the 
cell, we may or may not at some time discover. That she has con- 
scious aims is evident, from her adaptation of the form of the 
comb to circumstances, and from her ingenious contrivances, not 
only to repair mischief, but to guard against threatened evil. But 
it is equally evident that in the formation of the bee, and in the 
inspiration of her instincts, a knowledge and wisdom presided, to 
which the whole question of maximum and minimum lay open 
countless ages before human thought grappled with its problems. 

It only requires a more intimate acquaintance with the habits 
of any animal to discover the adaptation of its instincts to its or- 
ganization. The apparent instances to the contrary arise from 
want of patience and thoughtfulness in the observer. I stood, one 
evening, at early dusk, watching the movements of a curious msect 
on the inside wall of an open shed. Its body, a little over an inch 
in length, and very thin, seemed, nevertheless, too heavy for its 
long and delicate legs, which swayed and trembled under the 
weight, as it slowly stepped along, with long pauses between each 
step. It walked on four legs, holding the other two, which were 
shorter and stouter, extended in front. I presently perceived tliat 
it was making toward a fly which had settled, apparently to sleep, 
upon the board within three inches of my insect. I wished to see 
what its designs were upon the fly, but so slow were its motions, 
that I was obliged to wait fully twenty minutes before being grati- 
fied. As the insect approached the fly, he slowly extended a very 
long and exceedingly slender antenna, and touched the fly gently, 
in various parts, as if to ascertain more precisely its position. He 



42 GEOMETRY AND FAITH. 

« 

then made a detour, and brought himself, at length, exactly in 
front of the sleeping victim, with his own head nearly over the fly's 
head, and began very slowly to raise his raptorious legs high above 
the fly. I was growing tired of his slow and awkward motions, 
when, in an instant, my feelings were changed to those of the 
highest admiration for his great engineering skill ; the fly was 
aloft in air, with the beak of the insect thrust into its back calmly 
imbibing its juices ; while the fly's feet could touch nothing, its 
wings were both dislocated, and firmly pinioned in the captor's 
raptorious legs ; which, coming down suddenly between the wings, 
had parted them, dislocated them, and pinned them between the 
wrist-spurs of those legs and their sharp extremities ; then, with- 
out an instant's pause, lifted the fly from his feet, and impaled 
him upon the ploiaria's beak. 



IX. 

MOTION ETERNAL IN DURATION. 

The motion of bodies is not observed to be with uniform ve- 
locity. We see bodies at rest beginning to move, and bodies in 
motion coming to rest. Let us consider these cases a httle more 
closely. 

We have said that motion implies force, and that force implies 
will. Force is the energy of will acting upon matter. But how 
does the will affect matter which is foreign to will, and over which 
will, we might therefore suppose, would have no control ? To this 
question we answer that the human will never affects the material 
thing which it determines to move, except through the agency of 
material agents of whose existence and motions it may be uncon- 
scious. Not to speak of the unconsciousness of all earnest labor, 
the absorption of the mind in its object, take the case in which 
the mind is seeking to analyze its voUtions, and it Avill be found 
that we cannot reach, in our analysis, the first effect of the volition 
upon the physical frame. In the movements of my hand, although 
I know it is effected by the movement of certain muscles, and 
that this is effected through the nerves of volition, yet I cannot 
trace my will behind the command issued to the hand itself. 
Hence it may be said that the human will moves even the human 
body through physical agencies, and the manner of its control over 
these agencies is known only by Him who breathed into us the 

breath of life. 

^43) 



44 GEOMETRY AND FAITH. 

In like manner is it with the natural motions which we see on 
every side. All bodies are moved through the agency of other 
bodies, and we see nowhere a motion which is not dependent upon 
physical causes, that is, which is not produced by physical agents. 
Doubtless, He by whose will all things are moved is not restricted 
from any mode of action, and He can move bodies independently 
of all law, and without any intervention of means. Nevertheless, 
such motion would be miraculous, and out of the course of nature. 
Our will employs, unconsciously, the aid of nerve and muscle ; the 
Supreme Will employs, with wise designs, the intervention of the 
laws of impulse, attraction, and repulsion. 

But when a body at rest receives motion through impulse, it 
evidently continues the motion of the impelling body. So that, if 
the impelhng body is put to rest by its contact, the motion is not 
lost, but only transferred. And this is true independently of the 
size of the two bodies. The earth is not immovable, and the small- 
est grain of dust that falls upon it strikes with a certain amount of 
force. 

Again : when bodies act upon each other by attraction or repul- 
sion, the force acts upon both bodies. As surely as two vessels, 
floating on still water, would both move when one attempted to 
draw or push the other, so surely must each moving thing move 
those attracted by it, and all that attract it. 

This motion, also, exists, whatever be the relative size of the 
bodies. In the case of motion produced by direct attraction, — 
for instance, in the fall of bodies toward the earth, — the motion 
is in simple inverse proportion to the masses. It is, therefore, 
capable of easy arithmetical calculation. The weight of the earth 
in milligrams may be nearly represented by the figure 6 followed 
by thirty cyphers, or six nonillions. Hence the falling of a little 



MOTION ETERNAL IN DURATION. 45 

insect, weighing six milligrams, would move the earth the non- 
illionth part as much as the insect fell. That is, the thirtieth 
place of decimals is capable of representing the motion given to 
the earth by the fall of the smallest bodies. 

There is, then, in nature, no provision for the destruction of 
mution, but only for its transference. Motion in a body free from 
external influences is uniform in velocity and direction ; it can be 
retarded, and apparently destroyed, only by external influences, 
by impulse, or by attraction or repulsion. But these, we have 
shown, cannot really destroy ; they simply transfer the motion to 
the interfering body. Hence all motion is eternal ; it is communi- 
cated to an ever-increasing amount of matter ; it is, in each suc- 
cessive particle affected, less in quantity, but never becomes noth- 
ing, since the sum of the motion in all the particles remains the 
same. 

It is sometimes aflBrmed, since the demonstration of the theory 
that heat and light are undulatory motions, that all mechanical 
motions finally take the form of heat, or of light. The sound of a 
tuning-fork, placed upon a mass of caoutchouc, is inaudible, and 
a delicate ther mo-galvanic test shows that the temperature of the 
rubber has been raised. But a still more delicate test, if it were 
possible to apply one, might show a quaUty in this heat that would 
demonstrate it to have arisen from a fork ; just as Clairault's nicer 
calculation showed the comet of 1770 to have been, before it had 
its short period within the orbit of Jupiter, a wanderer outside the 
Jovian sphere. 



X. 

MOTION OMNIPRESENT IN SPACE. 

Chaeles Babbage, in the " Ninth Bridgewater Treatise," has 
a chapter concerning the permanent impression of our words upon 
the air, — a chapter which none have ever read without a thrill of 
mingled admiration and fear ; and which closes with an eloquence 
that were worthy the lips of an orator, though coming from a 
mathematician's pen. 

Would that Babbage had touched, in his fragmentary treatise, 
upon some of the inferences which may be drawn from the New- 
tonian law of gravity, — inferences which would probably have 
been as new to most of his readers as those which he, with so 
much acuteness, draws from the law of the equality of action and 
reaction. 

The motion of which Babbage speaks, in the chapter to wliich we 
refer, is undulatory, communicated by impulse, and requiring time 
for its transmission ; and the startling result of his reasoning comes 
from the never-dying character of the motion, keeping forever a 
record of our words in the atmosphere itself, always audible to a 
finer sense than ours ; reserved against the day of account, when, 
perchance, our own ears may be quickened to hear our own words 
yet ringing in the air. 

But motion is not only enduring through all time, it is simulta- 
neous throughout all space. The apple which falls from the tree is 
met by the earth : not half w^ay, but at a distance fitly proportioned 

(46) 



MOTION OMNIPRESENT IN SPACE. 47 

to their respective masses. The moon follows the movement of the 
earth with instant obedience, and the sun with prompt humility 
bends his course to theirs. The sister planets with their moons 
are moved by sympathy wdth earth, and the stars and most distant 
clusters of the universe obey the leading of the sun. Thus through- 
out all the fields of space, wherever stars or suns are scattered, 
they move for the falHng apple's sake. Nor is the motion slowly 
taken up. The moon waits for no tardy moving impulse from the 
earth, but instantly obeys. The speed of light which reaches the 
sun in a few minutes would be too slow to compare with this. 
Electricity itself, coursing round the earth a thousand times an 
hour, can give us no conception of the perfectly simultaneous 
motions of gravity. There are stars visible to the telescopic eye, 
whose light has been ages on its swift-winged course before it 
reached this distant part of space ; but they move in instant 
accordance wdth the falling fruit. 

True it is, that our senses refuse to bear witness to any motion 
other than the apple's fall, and our fingers tire if we attempt to 
write the long list of figures, which our Arabic notation requires 
to express the mov^ement thereby given to the sun. Yet that mo- 
tion can be proved to exist, and the algebraist's formula can rep- 
resent its quantity. The position of every particle of matter at 
every instant of time, past, present, or to come, has been written 
in one short sentence, which any man can read. And as each 
man can understand more or less of this formula of motion, accord- 
ing to his ability and his acquaintance with mathematical learning, 
80 we may conceive of intelligent beings, whose faculties are very 
far short of infinite perfection, who can read in that sentence tlie 
motions not only of the sun, but of all bodies which our senses 
reveal to us. Nay, if the mind of Newton has advanced in power 



48 GEOMETRY AND FAITH. 

since he entered heaven with a speed at all proportioned to his in- 
tellectual growth on earth, perhaps even he could now with great 
ease assign to every star in the wide universe of God the motion 
which it received from the fall of that apple which led him to his 
immortal discoveries. 

Every moving thing on the earth, from the least unto the great- 
est, is accompanied in motion by all the heavenly spheres. The 
rolling planets influence each other on their path, and each is influ- 
enced by the changes- on its surface. The starry systems, wheel- 
ing round their unknown centre, move in harmony with each other, 
and bend each other's courses, and each is moved by the planets 
which accompany it in its mighty dance. Thus does this law of 
gravitation bind all material bodies in one well-balanced system, 
wherein not one particle can move but all the uncounted series of 
worlds and suns must simultaneously move with it. 

Thus may every deed on earth be instantly known in the farthest 
star, whose light, traveling with almost unbounded speed since cre- 
ation's dawn, has not yet reached our eyes. It only needs in that 
star a sense quick enough to perceive the motion, infinitely too 
small for human sense, and an analysis far reaching enough to 
trace that motion to its cause. The cloud of witnesses that ever 
encompass this arena of our mortal life may need no near approach 
to earthly scenes, that they may scan our conduct. As they jour- 
ney from star to star, and roam through the unlimited glories of 
creation, they may read, in the motions of the heavens about them, 
the ever-faithful report of the deeds of men. 

This sympathetic movement of the planets, like the mechanical 
impulse given by our words to the air, is everduring. 

The astronomer, from the present motion of the comet, learns 
all its former path, traces it back on its long round of many years, 



MOTION OMNIPRESENT IN SPACE. 49 

shows you when and where it was disturbed in its course by plan- 
ets, and points to you the altered movement which it assumed from 
the interference of bodies unknown by any other means to human 
science. He needs only a more subtle analysis, and a wider grasp 
of mind, to do for the planets and the stars what he has done for 
the comet.. Nay, it were a task easily done by a spirit less than 
infinite to read in the present motion of any one star the past mo- 
ti'-ns of every star in the universe, and thus of every planet that 
wheels round those stars, and of every moving thing upon those 
planets. 

Thus considered, how strange a record does the star-gemmed 
vesture of the night present ! There, in the seemingly fixed order 
of those blazing sapphires, is a living dance, in whose mazy track 
is written the record of all the motions that ever men or nature 
made. Had we the skill to read it, we should there find written 
every deed of kindness, every deed of guilt, together with the fall 
of the landslide, the play of the fountain, the sporting of the lamb, 
and the waving of the grass. Nay, when we behold the suj er- 
human powers of calculation, exhibited sometimes by sickly chil- 
dren, long before they reach man's age, may we not believe that 
men, when hereafter freed from the load of this mortal clay, may 
be able in the movement of the planets or the sun to read the rec- 
ords of their own past life ? 

Thou, who hast raised thy hand to do a deed of wickedness, stay 
thine arm ! The universe will be witness of thine act, and bear 
an everlasting testimony against thee ; for every star in tlie re- 
motest heavens will move when thy hand moves, and all the tear- 
ful prayers thy soul can utter will never restore those moving orbs 
to the path from which thy deed has drawn them. 



XI. 

THE SPHERE OF HUMAN INFLUENCE. 

The conclusions of the last chapter would need but slight mocM- 
fication, should any future observations reveal the fact that the 
motions of gravity are not absolutely simultaneous. If gravity 
should prove to be a mere resultant of the undulations of light 
and heat, we should gain, indeed, a magnificent illustration of the 
inspired wisdom which begins its account of creation with recording 
the fiat, " Let there be light; " but we should not lose the spir- 
itual lessons drawn from the fact that the material universe is 
bound by gravitation into one sensitively-balanced whole, so that 
each deed of man is felt in the farthest star, and a perpetual rec- 
ord thereof is kept in the movement of the heavenly orbs. 

" Every natural fact is a symbol of some spiritual fact." As 
motion is propagated throughout all space, and endures through all 
time, so each change in the spirit of each man aifects the state of 
the spiritual universe, and its influence remains through all eter- 
nity. As matter by the law of gravity, so spirits by a law of sym- 
pathetic attraction are all bound in one harmonious whole, whereof 
" if one member suffer, all the members sufier with it." Liebig's 
law, that a moving particle communicates its motion to adjacent 
particles, was announced in defence of a mistaken theory of trichi- 
nosis, but the law itself is true, and universal in physics, in physi- 
ology, and in psychology. 

The law of attraction holds the same place of primary impor- 

(50) 



THE SPHERE OF HUMAN INFLUENCE. 51 

tance in considering man, as in considering matter. Our great 
economist makes the Unity of Law a fact of primal importance in 
the development of his grand and cheering theses. The eagle 
saint of the Christian Church declares that God is love ; and all 
the highest religions teach that man is made in God's image. He 
is the sun of infinite magnitude, the origin of all forces, but not 
moved by any reaction ; we are the particles moved by Him both 
immediately and mediately through each other. Love is the funda- 
mental law ; the sympathy between human souls is always greater 
than the antipathy ; even when, through disturbing forces, the 
sympathy is for a time neutralized, and the antipathy is developed 
into hatred. 

The influence which a man exerts does not cease with the effect 
that he has upon his most intimate friends ; nor does it flow from 
the power of his word alone, nor from the mere force of his 
example. Whatever a man does, or thinks or feels, even in soli- 
tude, has an effect upon the world. For, in the first place, it 
affects himself and his own character; and that character must 
influence, in some manner, those with whom he comes in contact ; 
influence them in proportion to the strength of his power to affect 
them, and to the weakness of their power to resist him. A cheer- 
ful countenance carries a gleam of sunshine into the darkest alley ; 
a sad face throws a shadow over the hearts of those who pass it, 
even on a crowded thoroughfare ; thus every shade of thouglit and 
feeling, expressed in the countenance, or in word, or gesture, or 
action, produces some corresponding change, slight thougli it may 
be, in all souls that recognize, however dimly, the expression. 
And this change transfers itself, in varying proportions, to ever- 
widening circles. Thus the spirit and tone of the a2;e is the sum 
of the individual thoughts, and thus also the individual character 



52 GEOMETRY AND FAITH. 

of each man is to some extent the product of all the preceding 
ages of the race. 

By the manners of a man, or by his speech, we know whether 
his companions have been Galileans or Athenians. A close ob- 
server in the city can tell, of the majority of strangers he may 
chance to notice, their age, their character, their calling, the 
place of their residence, and of their nativity, or that of their 
ancestors. It would only need a nicer observation, a closer in- 
sight, a more searching analysis, to detect in a stranger's heart 
both the original traits of his character and the modifications due 
to the influence of all with whom he has been associated. It might 
be a task no more above a Shakespeare's grasp, than the creation 
of a Hamlet is above the power of an ordinary man, to trace, in a 
man's present character, the influence of every person and every 
circumstance that have ever acted upon him to repress or to de- 
velop his powers. It would not require an absolutely infinite intel- 
lect to trace the effect of any humble act of an honest man, until 
it had been seen to have blessed millions of his fellow-men ; nor to 
show the loss or suffering that have flowed to thousands from an 
evil deed. It may hereafter be possible for some higher intelli- 
gence than ours to read the record of my interior life, written in 
positive or negative characters, upon the soul of some poor man 
whom I have never seen, but whom I must, nevertheless, have 
helped or hindered by my every act and word. 

Thus the spiritual universe is bound, by the law of love, under 
its wider enunciation of sympathetic attraction, into one finely- 
constituted whole, so that not one heart can throb but all hearts 
must throb with it. '' There is joy in the presence of the angels 
of God over one sinner that repenteth ; ' ' and to every man who 
falls into sin, we may say, with deeper meaning than that of the 



THE SPHERE OF HUMAN INFLUENCE. 53 

prophet, '' Hell from beneath is moved for thee, to meet thee at 
thy coming." 

As by the law of gravity the material universe, and by that 
of love the spiritual world, so by the association of ideas the 
world of thought is bound into one whole, whereof you cannot find 
one thought that is not connected, more or less directly, with all. 
Nothing known, nothing thought, nothing done, nothing felt, fails 
to leave a clew by which it may be recalled to memory. Each 
moment's state of consciousness is connected in a train which 
reaches back to the earliest moments of life, and shall reach on 
unbroken through eternity, so that it must ever be among the pos- 
sibihties of memory to recall the thoughts of any instant. And as 
the rare occurrence of unusual power, developed by accidental 
excitement, suggests hopes of an indefinite increase of power, 
when we shall have laid aside this frame, subject to accidents, so 
the preternatural manifestation of memory, in certain states of 
health, warns us that this possibiUty of recalling all things may 
become an actual reahty in the future Hfe. 

Then, as the soul surveys the past, with memory presenting its 
record of every thought and word and deed, and with an eye 
quickened to see the influence which each has had, she may sit in 
judgment on her own character ; and the word, wliich shall be 
the final judge, may speak through the soul itself. Then, also, as 
she enters the company of cherubim and seraphim, they will need 
no record of her good or evil deeds other than that written upon 
herself; from their eyes, as from her own, there shall, when she 
is present, be no past sin hidden, and no good thought concealed. 
In the anticipation of standing before such a tribunal, who can fail 
to find strength to resist the tempter, and encouragement in striv- 
ing after good ? 



XII. 

MAGNITUDE. 

Ix the minds of some who have read certain of the preceding 
chapters there has, doubtless, arisen an objection to accepting the 
results obtained, — the objection that the results are infinitesimal, 
and therefore non-existent. De minimis iion curat lex ; and that, 
it may be said, will be the opinion of the court of conscience con- 
cerning the '' permanent impression of our words upon the air ; " 
concerning the effect of our motions upon the distant stars ; con- 
cerning the influence of our character upon the tone of the spiritual 
universe. These influences will be infinitesimal, and to a large 
part will balance and destroy each other. 

In reply to these objections, I will begin by observing that the 
balancing of two forces is a real effect in nature, not to be for a 
moment confounded with the non-existence or destruction of the 
forces. If the earth do not rise to meet this falling rain-drop, it 
demonstrates that another drop, or its equivalent, is falling on the 
other side of the globe. 

Let me also concede, at once, that, in the form in which the 
arguments have been put, there is an assumption of certain laws 
of physics, which, being laws deduced from observation, may be 
subject to perturbations not yet discovered. Thus Babbage, in the 
chapter alluded to, assumes that the wave of sound runs friction- 
less through the air, the heat developed by the compression being 
absorbed in the expansion. But the experiments of Uriah Boyden 

(54) 



MAGNITUDE. 55 

have since shown that there is a slight amount of heat developed 
by the friction of the wave ; and this would slowly, but constantly, 
diminish the force employed in propagating the sound as sound. 
Again, I have assumed that gravity is a force acting at a distance, 
and requiring no time whatever for its transmission ; but it may 
possibly be hereafter shown that its speed is not thus absolutely 
infinite. 

With regard, however, to the main objection, that the infinitesi- 
mal may be neglected, the objection appears to me not valid, and 
to arise from the weakness of the human imagination. '- Time and 
space are great only with reference to the faculties of the beings 
which note them." In space and time themselves there is no 
natural unit, or scale of magnitude ; these are given only by 
thought, manifesting itself through material phenomena. Hence 
every scale of magnitude is relative to the mind employing it, and 
it is only the Unlimited Mind that can be free from the fetters 
imposed upon thought by the scale employed. 

In some of the nebulae we have examples of the spira mirahiUs 
of Bernouilli, drawn on a gigantic scale, so that the part visible to 
the telescope is many millions of leagues in extent. This spirnl 
may be drawn, on a smaller scale, by tracing upon a circumpolar 
map the path of a ship keeping steadily upon any one course, not 
to a cardinal point. Let us imagine this map extended until its 
radiating meridians stretch out among the stars, and let us trace 
upon it a spira mirahiUs^ which, beginning at a distance of one 
billion kilometers from the pole, is running thirty degrees north of 
east. The length of this spiral will be two billion kilometers, and 
it will make an innumerable number of revolutions in reaching the 
pole. Let us, next, imagine that we have come in upon the spiral, 
until we are but one kilometer from the pole ; the length of spiral 



66 GEOMETRY AND FAITH. 

yet remaining will be two kilometers, and the number of revolu- 
tions still to be made around the pole will remain innumerable. 
Let us again approach until we are within one millimeter from the 
pole ; the remaining length of spiral will be two millimeters ; but 
the number of revolutions yet to be made about the pole will still 
be innumerable. This little central part of the spiral, all lying 
within a circle two miUimeters in diameter, will be precisely simi- 
lar in shape to the whole spiral, two billion kilometers in diameter. 
If it were possible to look at this central part with a lens that 
should magnify a quadriUion times, it would appear precisely of 
the same size and shape as the whole spiral. By your approach to 
the centre, the scale alone would be altered ; the central part of 
the spiral would be a reduced picture of the whole ; its linear 
dimensions would be the fifteenth place of decimals of the dimen- 
sions of the original spiral. 

That is, it is the nature of the spira imrahilis^ that whether you 
move inward toward the pole, or outward away from it, the part 
between your position and the centre remains always exactly of 
the same shape, differing only in scale. Let us then approach to 
within the hundredth of a millimeter from the pole ; the central 
portion becomes now too small to be seen distinctly by the naked 
eye ; but it is of same shape as the whole ; and although its total 
length is only the fiftieth of a miUimeter, it still makes an infinite 
number of convolutions about the pole. On the other hand, if we 
should run out to the distance of a trillion kilometers, we should, 
probably, reach the distance of the furthest star visible to the 
naked eye. That is to say, the unaided eye enjoys a range of 
vision through about twenty places of decimals in linear dimension. 
And if we should revise Archimedes' calculation, on the number 
of sand-grains requisite to bury the universe, we should, conse- 



MAGNITUDE. 67 

quently, see that sixty places of figures will express the number of 
grains of the finest silt requisite to fill all space, out to the stars 
of the sixth magnitude. The sixtieth place of decimals is, there- 
fore, not zero ; it is the ratio of the space occupied by a minute 
grain of fine sand, to the space in which the fixed stars lie. In 
that minutest space the same forms may lie concealed as those 
which are illustrated in the whirlpool nebulae, and which might be 
conceived as filling vaster spaces. Twenty places of figures, in 
linear dimension, carry us beyond the limits of sight ; but space 
does not end there ; and we might affix figures forever, without 
arriving at a magnitude so great as to be impossible. 

Turning our thoughts again to the minute central portion of the 
spira mirahilis^ we could, by the aid of the microscope, see an 
object whose linear dimensions would stand in the seventh or 
eighth place of decimals of a meter. In that central speck, visible 
only to the best microscope, the wonderful spiral would still exist 
in all its perfection ; still making its proper angle with a line to 
the absolute centre ; still being in its fixed proportion of length 
to the length of that hne ; and still making an infinite number of 
convolutions around the pole. And this would hold could we, by 
the aid of more and more powerful lenses, continually approach 
the centre until our distance from it stood in the ninth or ninetieth, 
the nine hundreth or the nine millionth, place of decimals. We 
may write cyphers, after the decimal point, at the rate of nine 
million a second, for nine million centuries, but when we finally 
write a significant figure, that figure is not a cyi)her ; it is signifi- 
cant ; and if it signifies the distance which yet remains between 
our moving point and the pole which it is approaching, then, incon- 
ceivably small as the distance is, it has its relations : it is one-half 
the length of the remaining portion of the spiral ; and that portion 



58 GEOMETRY AND FAITH. 

of tlie spiral, nearly as it may be without any length or size, still 
makes its infinite number of convolutions about the pole. 

It has been inferred that, because the scale is thus capable of 
indefinite expansion and contraction, without any destruction of the 
form, or law of the curve, the contraction might proceed to reduce 
the spiral mto the absolute point at the centre ; and that afterward 
the point might be considered as an absolute nonentity, without 
destroying the spiral. Thus space and time, it has been said, may 
be shown to be purely subjective. 

But the failure in this attempt to demonstrate the subjective 
character of space is twofold. A point is not a nonentity ; it is a 
zero of magnitude ; yet it has position, or is a position. It is not 
in the mind, it is in space, fixed in its position, although without 
magnitude. But although thus real, and indestructible in space, 
yet being without dimensions or parts, it is incapable of containing 
a curve except by a figure of speech ; by which we either attribute 
to the point the potentiality of the subjective law ; or else speak 
of a point when we simply mean an infinitesimal space. As we 
diminish, for example, the scale of the spira mirahilis^ — by run- 
ning in upon it towards the centre, and considering only the part 
yet remaining, which is always similar to the original whole, — 
it remains real, so long as we have not arrived at the actual 
centre ; but, when that point is reached, the spiral has vanished ; 
there is nothing remaining between us and the pole, for we are at 
the pole. That pole does not then become subjective ; it is a real 
point ; but it does not contain the spiral, except potentially. 

In dealing with the infinitesimal and the infinite, the practice 
of geometers varies ; some delight in stating truths in forms which 
seem false and self-contradictory ; others carefully avoid such 
forms. Both classes are liable to error ; the impossibility of 



MAGNITUDE. 59 

clearly imagining the infinite and the infinitesimal is not destroyed 
by any forms of language ; and the difficulty of reaching true con- 
clusions is not necessarily increased by the use of forms of speech 
concerning them, literally incorrect. The calculus of Leibnitz is, 
in a majority of its propositions, Uterally false ; yet it always leads 
to true conclusions, in the hands of one who can distinguish 
between the letter and the spirit. 

The best geometer can, however, err ; as, for example, some 
have said that the point approaching the pole, in a spira mirahilis^ 
can never reach the pole, because it will always have an infinite 
number of revolutions to make before reaching it. The premise 
is true, but the conclusion false. The point will always, until it 
reaches the pole, have an infinite number of revolutions to make 
before reaching it. But if the motion in the spiral be with uniform 
velocity, then the revolutions will be with increasing velocity ; and 
finally with infinite velocity ; so that the infinite number of revolu- 
tions will be accomplished in a finite time. To take the particular 
example we have been considering, — the point moving at a con- 
stant angle of sixty degrees with the line to the pole, — if the 
point moves uniformly at the rate of a slow walk, say four kilo- 
meters an hour, then it will reach the pole in precisely one hour 
from the time that its polar distance is two kilometers. But, while 
the motion of the point is uniform, at four kilometers an hour, and 
its approach to the pole uniform, at half that rate, the angular 
velocity constantly increases, being always such that (if it could 
remain uniform) it would carry the point around the pole in about 
three and five-eighths the time that remains of the hour. Thus, 
when at the distance of a meter from the pole, the revolution 
would carry it around nine times a minute ; while at the distance 
of a millimeter the revolution would be at the rate of one hundred 



60 GEOMETEY AND FAITH. 

and fifty circuits a second ; the rotation thus increasing in rapidity, 
in direct proportion to the approach to the pole. But in less than 
the one five-hundredth of a second the hour has expired ; the centre 
is reached ; the whole spiral has been passed over, and the point, 
having made innumerable revolutions, is at the pole. The pole 
does not contain the spiral ; although it may be said to contain it, 
in the sense that the spiral, with an infinite number of coils, is con- 
tained in any portion of space around the pole, however small, even 
if smaller than any portion that can be measured, named, or im- 
agined. Thus the point may be said to contain the spiral poten- 
tially, so that the spiral would become actual, could the point be 
magnified. 

This illustration of the logarithmic spiral has been chosen and 
fully expanded, not only because of its peculiar adaptation to the 
illustration of similarity, and its curious property of always having, 
even when reduced to an infinitesimal size, an infinite number of 
coils, but for the historic associations with that curve which Ber- 
nouilli would fain have had carved upon his tombstone. The same 
conclusions would be reached did we consider the shrinking of any 
other forms in just proportion. It is conceivable that the entire 
universe might be altered in size, and if proportional changes went 
on, in every part, in all the forces acting upon it, and in the facul- 
ties of all creatures, it would not be in the power of human beings 
to discover that the change was made. Were the universe thus 
reduced to the twentieth place of decimals in linear dimensions, 
and the requisite changes made in the forces of nature and in the 
faculties of man, the whole stellar system would be contained in 
the space now occupied by a grain of sand. A second reduction 
might take place, and a third, and so on forever ; and, unless the 
rate at which the changes were made increased, the universe would 



MAGNITUDE. 61 

8till remain to its inhabitants as large and grand as before. But 
let the rate increase so as to bring the universe to a point, and its 
actuahty would be gone, and its potentiality alone remain. Still 
that potentiahty would be objective to the Creative Mind ; objec- 
tive in the point. Sweep the point away, and the universe would 
exist only, as in Erigena's " Second Division of Nature," subjec- 
tively in the Divine Thought. 

We thus reach sublimer conceptions of the immeasurable gulf 
between the human and the Divine Mind by holding to the 
veracity of that intuition which pronounces space and time inde- 
structible objective entities ; and renouncing the philosophic wis- 
dom, made popular by Teufelsdreck, which accounts them mere 
modes of human thought. To the human mind there is no unit 
of space or of time, save those given in the creation ; we cannot 
even imagine any unit not thus given. To the Divine Mind alone 
belongs the possibiHty of deciding on the scale of creation, and 
deciding what men shall consider large or small, brief or lasting. 
As the human eye requires the aid of the telescope at one end of 
its range of vision, and the microscope at the other, so all our fac- 
ulties are limited to that which is neither too great nor too small 
for us. But, in regard to the works of nature, we neither discover 
limits, nor are we compelled by any mental necessity to suppose 
that there are limits. The bounds of the universe are independent 
of the weakness and limitations of our powers of imagination. 

There is, therefore, no impossibility in the specuhitions of Lov- 
ering, published in the " Cambridge Miscellany '' in 1842; that 
the atoms of our universe may be stars and suns of a smaller one, 
composed in like manner of infinitely smaller stars and suns ; while 
our constellations and solar systems may constitute only molecules 
in a vaster world. The infinity of space would almost seem to 



62 GEOMETEY AXD FAITH. 

demand such an arrangement to utilize its wastes. The human 
mind, fettered by the body, seems in such speculations to show its 
kindred to the Infinite Spirit, to. whom 

" There's nothing great appears, 
. . . There's nothing small." 

And these speculations are not confined to a few learned men, 
whose studies lead them naturally to the theme. Fifteen years 
before the publication of Lovering's paper, I myself, a child, heard 
other children supposing that this universe might be the atoms in a 
crumb let fall by a giant ; and that the slow precession of the 
equinoxes might be the rotation of that crumb, in the air, as it fell 
to the ground, in an infinitely larger universe. 

The true greatness of a work is in the thought which it embodies, 
not in the scale on which it is wrought. The idea or law of the 
spira mirahilis has a fascinating beauty to a geometer ; but to such 
a mind it is a matter of perfect indifference whether that spiral be 
illustrated in the minutest shell or in the largest nebula. I have 
been sometimes as much moved at a slight wood-cut outline of a 
mountain range as at the sight of the vast masses themselves. The 
Dead March from Saul will express grief on a grand scale, a sense 
of human weakness resting in unshaken confidence on the Divine 
strength, whether played on a single instrument or with a full 
orchestra. 

Our speculations on the scale of magnitude become, therefore, 
unimportant. Whether there are, or are not, infinitesimal worldi 
included in the atoms of our worlds, and infinite worlds in whicL 
our systems are atoms, the grandest and most inspiring object for 
our contemplation is the law, plan, or thought, on which the uni- 
verse, within reach of our faculties, is built. We need not reduce 



MAGNITUDE. 63 

it to a point, or dissolve it in ideal subjectivity, in order to show 
its unity. The universaUty of gravity, the co-extensive universality 
of Hght, which by the spectroscope has shown that some, at least? 
of our chemical elements are universally dififused, the correlation 
of forces, the adaptation of all parts of the universe to each other, 
all tend to conifirm the sublime conclusion that the universe is the 
expression of one infinitely complex, yet infinitely simple, connected 
thought, in which all was foreseen, and all comprehended at a sin- 
gle glance by the Intelligence which framed it. The aim of science 
is to develop and trace the connection of the parts of this intel- 
lectual whole ; the end of religion is to interpret for the heart 
and soul the lessons given through this intellectual form. 



XIII. 

CHANCE AND AVERAGE. 

When two phenomena arise from entirely independent causes, 
the relation of one to the other is said to result from chance. The 
disposition to consider chance an actually existing cause is so 
great, that men have, in all ages, personified, and in some nations 
even deified it. 

In the highest contemplation of the universe, as the realization 
of one grand conception of the Divine Mind, it might be thought 
that the idea of chance would be excluded, because all phenomena 
would then be regarded as springing from a single cause ; all the 
minutest events would be considered not only as foreseen, but as 
intended ; as the necessary results of the original thought made 
actual in the universe. 

But the idea of chance, of relations in events springing from 
independent causes, is so positive in its character, that we are 
unwilling to concede it to be a mere result of the weakness of the 
human mind, of our inabihty to rise to a habitual contemplation of 
one First Cause. It seems more like a direct gift of power ; a 
power to apprehend some really occurring phenomenon in nature. 
As such, it forms the basis of a distinct and valuable calculus, 
applicable to important economic questions of assurance and annui- 
ties, and to weighty scientific problems, as a test of hypotheses, 
and a criterion for rejecting doubtful observations. The successful 
application of this calculus of probabilities to so many actual prob- 

(64) 



CHANCE AND AVERAGE. 65 

lems in the universe is a demonstration that, however diiScult it 
may be to reconcile the conception with our ideas of " foreknowl- 
edge absolute " in the single Creative Will, we must, nevertheless, 
admit into our theory of the world the conception of independent 
causes, leading to what may be justly called accidental results. 

The reconciliation of this contradiction, so far as reconciliation is 
possible in our finite minds, is probably to be found in the con- 
sideration of averages. In our human work we frequently act 
upon a multitude of individual objects, without special designs in 
regard to each, but with a general regard to the average action 
and to the total result. The sower does not consciously choose 
the position in which a single grain of his wheat shall fall, yet 
designs and accomplishes an even cast of a given quantity of seed 
to the acre. The causes which determine the position of each 
grain are so numerous, and their connection so remote, that they 
may be considered, for one grain, independent of those for another. 
In throwing a die repeatedly, in like manner, the causes which 
determine its position after one throw are so numerous, and so re- 
motely connected with those that determine its next position, that 
they may be considered independent. Yet the throws are so 
governed by our will that we may decide, beforehand, how many 
to make in each minute, — and the positions are so determined by 
the shape and material of the die, that if it be a homogeneous 
cube the tendency will be, as the throws are multiplied, to have 
each side uppermost one-sixth of the time. 

This is the law of chance, as applied to averages. And as 
chance has been personified, and even deified, so average has, by 
some writers, had divine powers ascribed to it. It has been gravely 
asserted that the saving of a man from criminal courses only drives 
another man into crime to keep up the average ; as though the 



66 GEOMETRY AND FAITH. 

present average had an inherent power to perpetuate itself; as 
though dice could not be loaded without producing a counter- 
loading in the other player's dice ; as though the sower could not 
vary the size or cast of one handful without immediately varying 
another handful to keep his field from having more or less seed 
upon its surface. 

The average is a result, not a cause. It is the result of rela- 
tions that exist between the various causes producing the eJSects, 
and may be changed at any time by interfering between those 
relations ; by the dice being loaded, or the sower walking at a dif- 
ferent pace ; by Jenner's introduction of vaccination, or the dis- 
covery of America putting quinine into the physician's hand ; or 
John Howard visiting the prisons, or the Apostle Paul receiving 
his commission. Great changes thus take place in natural aver- 
ages ; and small changes may be made at any moment by the 
action of the human will. 

It seems not unworthy oar highest conceptions of the divine plan 
to suppose that certain groups of phenomena in nature may, like 
the sowing of seed by man, be directed and intended for average 
results, without special design for each individual case. The im- 
passable gulf between the finite and the Infinite Mind would still 
remain, in the ability of the Divine Providence to select at will 
any one of the innumerable cases, and employ it as a means to 
higher and further ends. The winds from the Mediterranean, for 
example, bring a fixed average of vapor to the summits of the 
Alps, where it is showered down in countless crystals of snow. 
These, under slight changes of temperature, contract into minute 
globes ; and these particles of ice, piled up in the mountain bashis, 
press themselves into an almost solid mass, and push themselves, 
or their earlier companions, down the valleys, grinding ofi* the 



CHANCE AND AVERAGE. 67 

rocks into powder, which is washed by the melting ice into the 
rivers and into the sea. The magnitude of these glaciers is limited 
by the quantity of snow ; and, in its turn, limits the quantity of 
gravel and sand, and the size of the boulders formed by them. 
To these results, a thousand causes which I have not mentioned 
conspire. A theist, believing that the glacial system of the mod- 
ern Alps is part of a divine plan, is not obliged to suppose that 
this plan includes the position of every individual grain of silt, in 
the ocean bed, brought from the Alps ; is not obliged to make this 
supposition, even if to his theism he adds faith in a special Provi- 
dence, and thinks that a grain of sand may be a providential 
instrument in effecting some great result. 

But whatever our explanation of the occurrence of so-called 
chance among the averages of nature, these chances and averages 
are frequently adapted to each other with a harmony that seems 
to admit of no other solution than a reference to the Divine plan 
which fits each to all and all to each. Enthusiastic students of the 
calculus of probabihties sometimes represent all human judgments 
as the result of a calculation of chances ; and our certainties are 
said by them to be merely propositions, the truth of which is in- 
finitely probable. Many of the arguments of natural theology, so 
called, can be very conveniently put into this form. In the forma- 
tion of planets around the sun, according to the nebular hypothesis, 
the chances were small against an order which should fail to pre- 
serve the stability of the system ; and the present harmony of dis- 
tances must be referred, directly or indirectly, to presiding thought. 
In the formation of the solar system, the chances were small that 
this particular planet should have its elements mingled in ])rccisoly 
that proportion which has resulted in so full a development of life 
and of human activity ; and the arguments of Prof. Cooke's '* Ro« 



68 GEOMETRY AND FAITH. 

ligion and Chemistry" derive from this consideration a demon- 
strative force. 

In the course of this successive development of vegetable and 
animal life upon the earth, there has been, with frequent muta- 
tion, a general permanence. Scientific speculation at the present 
time busies itself with the question whether the permanence has 
been real, and the changes sudden ; or whether the stability is 
seeming, and the mutations have always been going on with 
stealthy step. Whichever of these theories proves to be true, 
that of Plato, or that of Democritus, there is a seeming stability 
in the present species, which have lasted without sensible change, 
except the extinction of some kinds, for thousands of years. Ac- 
cording to the theory of Democritus, as revived in our days, this 
arises from the fact that the species at present existing are the 
fittest for the existing epoch, and thus survive. According to the 
rival theory this fitness arose from no blind struggle for life, but in 
accordance with a Divine plan, fulfilled by divine power. In ordi- 
nary cases the judgment may possibly remain suspended, whether 
to suppose the Divine Will acted in reference to the perpetuation 
of a species by some general law, covering many species, or by 
special adaptations to one. These cases may therefore be dis- 
missed from the argument. If we grant that a blind evolution by 
natural variation and survival of the fittest will explain them, it 
must also be conceded that an intelligent adaptation of the organ- 
ism to its medium will also explain them. But there are other 
cases, in which the imagination runs riot in vain for any " suffi- 
cient reason ' ' outside of the will and purpose of the Creator, ful- 
filling an original plan. These cases are like " experimenta cru- 
cis," — the theory that fails to hint at a possible explanation fails 
to explain the universe. 



CHANCE AND AVERAGE. 69 

Such, it appears to me, are the cases in which the fecundity of 
a creature is in inverse proportion to its chances of life. I would 
by no means say that these are the only points in the animal and 
vegetable economy which the evolution theories appear to me to be 
utterly incapable of explaining ; but they are the cases w^hich fall 
under the head of average and chances, and demonstrate that the 
Eternal Thought which planned this present world comprehended 
all and more than all which is included in our modern calculus of 
probabihties. 

If the ovaries of the Dodo contained one thousand ova, and if, 
on an average, less than one of these grew into an adult female 
Dodo, with equal chances of propagating her kind, it is evident 
that the Dodo must become extinct. If, on the other hand, two 
of these ova were impregnated, and came to maturity, it would 
take but a few generations of the bird to cover the earth, and 
exclude all other beings. This is prevented, it may be said, by 
the struggle for life. But the fecundity of each species must be 
exactly proportioned to the chances of failure in that struggle. 

The horse-hair eel is said to lay several millions of ei^gs ; let us 
say five million. Why this enormous fecundity ? Because the 
chances of the eggs coming to maturity, as eels, is so small. In 
order to keep the species in existence, two in five millions (if the 
sexes are of equal numbers) must succeed in escaping all the 
dangers which beset the eggs and the young in the brook, and 
then succeed in finding, near the brook, crickets or grasshoppers 
into which they may penetrate. These grasshoppers must escape 
their enemies, and survive the depredations of the hair eels, until 
the latter reach maturity, when they must escape near enough to 
a brook to find their way there, and meet hair eels of the opposite 
Bex. The chances are two in live millions, let us say, and the 



70 GEOMETRY AND FAITH. 

creature lays five millions of eggs. Did she average but four 
millions, the race would in a few years become extinct ; did she 
average six, the creatures would multiply in a few years beyond 
all bounds. The permanence of the species for so many years 
demonstrates the accuracy with which its fecundity is proportioned 
to the slimness of its chance in the falsely-called struggle for life. 



XIV. 

PHYLLOTAXIS. 

The reader may be interested in a more detailed development 
of the arguments briefly alluded to upon pages 5 and 6, 35 and 36. 

There is a rule in arithmetic called the rule of False, or the 
rule of Position. It is the most general, and the most useful of 
all rules in the art of computation. Its method renders it appli- 
cable to every problem in which the accuracy of an answer can 
be tested. The rule may be briefly stated as follows : Guess at an 
answer, and test by numerical computation the accuracy of its 
results. If the results are accurate, the answer stands the test 
and was correct. And, if the results are not accurate, the error 
affords data for estimating the error of the assumed answer ; and 
making a second better guess, to be tested as the first was ; and 
thus to afford data for a third guess, still nearer the truth. 

This method of hypothesis and verification is applicable not 
only to arithmetical problems but to all questions of j)ractical 
science. It may even be said to be the general rule by which tlie 
finite mind approaches every trutli which it can aj)proacli ; and 
thus reaches every truth wliich it can reach. In some minds ilie 
process is rapid and with a very obscure and evanescent con- 
sciousness of its operation; in others slower and with distinct 
knowledge of the steps ; and in all minds tlie ra])i(lity and ease 
of the process vary with the nature of the problem. I hit in 
regard to all ])henomena, which require a cause or theory to 
account for them, we are not able to draw tlie theory out o{ the 
facts; the theory comes from our own minds; we i)lace it among 

71 



72 GEOMETRY AND FAITH. 

the facts; and then adopt, modify or reject it, according to its 
agreement with the facts and ability to explain them. Some- 
times the theory explains the facts perfectly, but goes no further; 
as, for example, when we assume a centre for a circumference 
passing through three points ; the final verification of the centre, 
by proving it equidistant from the points, gives us nothing fur- 
ther to do ; we have found the centre, that is all. But, in more 
complex cases, it frequently happens that the theory proposed 
and tested, for the explanation of one set of facts, proves to be 
also a satisfactory explanation of other facts ; its value and 
authority as a theory are thereby immediately greatly enhanced. 
Nor is it in the physical sciences alone that this increase of cer- 
tainty and value may be found ; the operations of the intellect are 
similar, on whatever class of objects it is operating; and theology 
itself may be lawfully treated, to a certain extent, as Spinoza 
treated it, by mathematical methods. 

Thus Maupertuis assumed it as a fundamental principle that 
the Divine Being, being unerringly wise, would waste no energy ; 
that everything in nature must therefore be done with perfect 
economy of force. This theological dogma is called, in mechanics, 
the principle of the least action ; and is, in that science, an invalu- 
able and fruitful principle; as may be readily shown even to 
those not skilled in mathematical analysis. Take for example the 
mechanical theory of light. How shall light move in a uniform 
medium, according to this principle of the least action? Evi- 
dently in a straight line, since that is the shortest distance between 
two points. How shall light be reflected from a polished surface ? 
Evidently so as to make the sum of the incident and reflected 
rays the shortest possible in going from the point giving light to 
the point on which the reflected ray shines ; and a simple calcu- 
lation shows that this requires the incident and reflected ray to 
make the same angle with the reflecting surface. How shall light 



PHYLLOTAXIS. 73 

be refracted in passing from one medium to another? Here, 
again, we must have the sum of the incident and refracted ray 
such as to make the whole power exi3ended as small as possible ; 
and a simple calculation proves that this requires the sines of the 
angles of incidence and reflection to be in proportion of the dif- 
ficulty of moving in the two media. These instances show the 
usefulness of the principle of least action in physical inquiry. These 
results concerning light are amply confirmed by experiment ; and 
shown to give the manner in which a ray of light actually behaves. 
But when we turn the argument about, and from this agree- 
ment of the experimental results of observations on light, with 
the principle of the least action, would argue that the creator of 
light uses a perfect economy of force, and is therefore unerring 
in wisdom ; we feel at once that the argument is not absolutely 
conclusive. It creates a presumption, but does not force a con- 
viction. We see that the motion of light in a straight line, reflec- 
tion at an equal angle, and refraction by Snell's law of sines, 
involve perfect economy of force ; and yet these may be neces- 
sary results of the constitution of the luminiferous ether, not 
foreseen when the ether was constituted. In order to make the 
argument from morphology or from teleology conclusive, the 
instances of the divine thought and forethought must be such as 
cannot be explained from mechanical or mathematical necessities. 
The fact that Maupertuis's theological doctrine of the divine 
economy of force has led, or could have led, to such numerous 
discoveries of the actual laws of physics, certainly creates a 
strong presumption in favor of its theological truth. It shows 
that Bacon's sneer at barren virgins consecrated to God is wholly 
uncalled for; here is a religious tenet, a purely theological doc- 
trine actually giving to mathematicians and ])hysicisls more 
knowledge of the universe than they could get from observation 
of nature, without its aid. 



74 GEOMETRY AND FAITH. 

There are various other points in which this theological hypothe- 
sis of an infinite wisdom directing the world can be subjected to 
tests ; and in some of them it stands the test, so perfectly as to 
rise rapidly toward a settled and demonstrated theory; even 
upon these lower grounds of the understanding, and independent 
of the intuitions of the higher reason from which Maupertuis and 
the theologians announce it. Let me briefly touch upon two of 
them. 

In the early history of astronomy it w^as assumed, that the law 
of the movement of the heavenly bodies must be a perfect law. 
The planet, swinging free in space, is subject to no interruption 
from finite hindrances, but moves under the influence of universal 
law^ alone ; therefore its motion is perfect. This grand theologi- 
cal conception is worthy a place beside Maupertuis's principle 
of the least action. But, in attempting to verify this hypothesis, 
the ancient astronomers made another assumption not so fortu- 
nate and trustworthy; the assumption that circular motion is the 
only perfect motion. They assumed that the orbit of a planet is 
circular. But on putting this to the test it failed to account for 
the appearances in the heavens. Still this did not shake their 
faith in the perfection of circular motion. They only thought 
that there must be a combination of circles. Instead of a single 
arm carrying the planet, they put an arm upon the end of an 
arm ; and conceived it as rotating twice, while the first arm was 
rotating once. If the second arm be very short, in comparison 
with the first, the j)ath of the planet carried by it would differ 
but slightly from a circle ; while if the arms are nearly equal in 
length the path would be very different. But they soon found 
that no imaginable proportion between the lengths of the two 
arms would enable them to represent well the planetary motions. 
Still adhering to the circular movement, they place a thiYcJ arm 
at the end of the second. This third arm was to revolve three 



PHYLLOT.IXIS. 75 

times, while the second revolved twice, and the first arm once. 
By giving proper proportions to the lengths of these three arms 
the orbit of any planer could be described with tolerable accu- 
racy. By the addition of fourtli and fifth arms, to revolve re- 
spectively four and ^ve times, while the first revolved once, the 
positions of the planets for a single revolution in their orbits 
could be given with the utmost nicety. 

These epicycles of Hipparchus are no longer used in astron- 
omy ; but they have been used in modern days, in other dej»art- 
ments ; in what might be called the statistics of physics, — 
tabulated observations. It has been shown that with from three 
to six arms, rotating each with a rapidity proportioned to the 
number of the arm, beginning at the centre ; the end of the outer 
arm may be made to move in any path required, provided we 
may fix the length of the arms as we choose, and put them in 
what position we choose, at the beginning of the motion. And 
it is justly esteemed a grand work of Hipparclius and of modern 
analvsts (including our own Peirce) that they sliould liave invented 
and perfected by two thousand years of study so remarkable a 
result, as that of describing any outline whatever, by the simj^le 
device of rotating, simultaneously, three or more radii linked 
together by their ends. Yet in nature a similar device had been 
used in a rude form at least from the eozoic ages ; and in a per- 
fected form, even better than that of Hipparchus and incomj>ara- 
bly more practical and rich in results, from the very advent of 
man upon the planet. The difference between the epicycles of 
Ilipi^archus and those of nature, is that in the former, the ratio 
of the rapidities of rotation is fixed, and the ratio of the lengths 
of the rotating arms is varied at pleasure; while in nature it is 
the pro]>ortion of their lengths that is fixed, and the i>roportion- 
ate velocity of rotation is varied at pleasure. For what is this 
riijrht arm and hand of man but a linked series of radii in which, 



76 GEOMETRY AND FAITH. 

from the mechanical necessities of the skeleton, the length of the 
spokes is fixed, but the resulting stiffness is much more than 
compensated by the variable proportions allow^ed to the veloci- 
ties of rotation. The end, free motion in any direction, could of 
course be attained without a skeleton ; as for example the tip of 
the human tongue can readily be trained to move in any conceiv- 
able path within the buccal chamber. But when strength and 
dignity have been given to the frame by a skeleton, and mobility 
by articulations, the articulations are exquisitely adapted to the 
psychic needs of the species. No mechanical necessity can be 
conceived as producing this adaptation ; nor can I see the proba- 
bility, or even possibility, of this adaptation having been pro- 
duced by the mutual reaction of the psyche of the animal and its 
environment. 

In the case of man, his immensely varied mental and spiritual 
powers and capacities ; his need of actions incalculably more 
varied than those needed by the brute ; whether we regard his 
movements as aimed to jDroduce physical or mental effects ; require 
that he should have much greater freedom of movement under 
definite control. And this is accomplished for him, partly through 
the perfection of his general form, so " express and admirable ; " 
but more particularly by the epicycloidal movements of the hand. 
The shoulder is the centre of these movements, but it is not 
rigidly fixed, it has a proper motion of its own by the sway and 
torsion of the trunk, by the movement also of the lower limbs, 
by which the whole body has a capability of motion. But assum- 
ing, for the moment, this moveable shoulder-blade to contain in 
its socket the centre of the epicycloidal movement of the hands; 
we have first the humerus, swinging freely in all directions, 
within a cone of about 120° ; which is accomplished by what is 
called a ball and socket joint. The second link consists of two 
bones in the fore arm, hinged to the first link by a hinge joint; 



PHYLLOTAXIS. 77 

but this joint receives an equivalent for ball and socket freedom, 
by the humerus rotating about 90° on its own axis. The third 
link is the metacarpus, jointed to the arm by a compound hinge, 
which, however, has an equivalent for ball and socket freedom by 
the peculiar semi-revolution of. the tw^o bones of the forearm 
about each other. For ordinary movements these three links 
suffice ; the phalanges being kept in a fixed position with reference 
to the metacarpus ; just as three arms suffice for the easier curves 
in the Hipparchian epicycles. But for the nicest delicacy there 
are the three joints of the phalanges, i^erf acting the instrument, 
just as additional short radii are sometimes needed to perfect the 
epicycles. 

Every one is familiar with the fact that a trained hand can 
sweep, with a crayon, any outline or figure whatever upon the 
blackboard. What I am endeavoring to show^ is, that this ability 
is furnished by an adaptation of the skeleton and muscular system 
to freedom of motion, and to freedom under control ; by a system 
similar to the Hipparchian epicycles, famous in astronomy and 
physics. In fact, every motion of the finger ends, so long as tlie 
feet remain in one spot, is an epicycloidal motion; the multitude 
of links giving it a practical infinity of possible forms. Even the 
six links from the scapula to the finger-ends give an incalculable 
variety of possible forms, which our imagination cannot distin- 
guish from an infinite variety. Here then is a wisdom and skill in 
the adaptation of the body to the mind of man, which seems to 
me an irrefragable proof of the existence of a wise designer of the 
animal kingdom. 

Let us turn now to the vegetable kingdom for the second in- 
stance by which I would test the hypothesis of the world being the 
work of an infinitely wise Builder. Suppose such a Buildi'r t(^ 
design a part of the vegetal)le kingdom to grow by leaves and 
buds on an ascending axis; the leaves to be the lungs and stomach 



78 GEOMETRY AND FAITH. 

of the plants, and to require light for the fulfilment of their 
functions. In the usual growth of such 23lants it will be manifest 
that lio'ht from the zenith will be most valuable. Side lio^hts will 
be more apt to be cut off by neighboring plants or other obstacles ; 
and, even if not cut off, may have and will be likely to have the 
highest pitched and most valuable vibrations absorbed and 
destroyed or diverted by passing horizontally through so much 
more of the lower atmosphere. The first necessity of a leaf, 
therefore, will be zenith light. And it will be expected of uner- 
ring wisdom, we might almost say it would be expected of the 
divine justice, that each leaf should have the fairest possible 
chance to have zenith light. In other words, it will be ex23ected 
that the leaves of plants should not be placed one over another ; 
but they should be scattered around the stem in such way as to 
give each the least shade, and the most light possible. Of course 
it will not be essential that this should be done with perfect ac- 
curacy ; practical ends do not need theoretical exactness ; on the 
contrary, it is a mark of a poor workman to have him give a 
degree of finish and exactness incommensurate with the nature of 
the work ; finishing, for example a kitchen clock as one would 
finish a clock for an astronomical observatory ; that is waste, not 
economy, of power. But we should expect in plants built by an 
infinitely wise creator to find distinct evidence that a general 
plan, for the accurate distribution of the leaves around a vertical 
stem, was in operation ; not distributing them in exact conformity 
with the plan, but near enough for practical purposes ; and evi- 
dently showing a perfect knowledge of the perfect plan. 

But Avhat would be a perfect plan ? Like the old astronomers, 
we are assuming that the plan of creation is perfect; let us not too 
hastily assume, as they did, that we know what a perfect jDlan 
would be. But, as near as we can see it, a theoretically perfect 
plan ought to be a symmetrical plan ; to go on by a general law ; 



PHYLLOTAXIS. 79 

and to place the leaves always as far apart laterally as possible, as 
we go up the stem ; so that they shall shade each other as little as 
possible. The first leaf being at the bottom, the second one going 
up must be nearly opposite, and thus divide the circumference of 
the stem nearly in halves. Yet when the third leaf comes out still 
higher, the circumference, as you look down upon the plant, must 
be nearly divided into tliirds. When the fourth leaf is added, the 
circumference must be nearly divided into quarters ; when the 
fifth leaf conies out, nearly into fifths ; and so on. The distance 
round the stem, the angular distance, as you look straight down 
on the end of the stem, between any leaf and the one next above, 
or next below it, must evidently be between ys and ^ a circum- 
ference ; or else between ^ and 2^ ; were it not so, the first three 
leaves could not fulfil the requirements of the problem. 

These requirements with reference to tlie first three leaves can 
easily be put in a simple algebraical equation. The second leaf 
must divide the circumference in halves as nearly as the third leaf 
makes thirds of it. That is to say : The remainder of the cir- 
cumference, after two angles have been taken out, must be to tlie 
remainder after only one was taken, as that one was to tlie 
whole. But by a well-known proposition in ju'oportion this yields 
at once the proportion that the angle is to the remainder of the 
circumference, as that remainder is to the whole circumference. 
In other words the circumference should be divided in extreme 
and mean ratio; a curious expression familiar in geometry, signi- 
fying the division of a thing into two parts, such that the smaller 
part is to the larger as the larger is to the whole.* Division in 
extreme and mean ratio was invented by the early geometers, 
before the Christian era, as a means of inscribing a five-sided 
figure in a circle ; but it was never suspected that it occurred in 

* Let the arc AB be x ; the circumference, 1. Let BC aiul CD each = AB = x. That 
the halves AB, and BOA, may be in the same proportion as the thinls AB. BC, 



80 GEOMETEY AXD FAITH. 

nature until 1849 years after that era. The smaller part is nearly 
382 thousandths of the whole ; and, if any one wishes to calculate 
it more nicely, he may extract the square root of 5 as far as he 
pleases, subtract it from 3, and divide the remainder by 2. It is 
a peculiar fraction ; added to its o^\ti square root it produces 
unity ; it is also one-third of the sum of 1 added to its square. 
But, as the square root of 5 is inexpressible in numbers, this pecu- 
liar fraction is also inexpressible. Its presence in nature, were it 
there, could not be detected by any microscope. All that we 
could hope to find would be approximations to it. If a j^lant had 
such a law of growth that the successive leaves had a tendency to 
follow each other at equal intervals, each .381966 of a circumfer- 
ence horizontally from its neighbor, we could not measure the 
angle from centre to centre of the leaves, accurately to any thing 
like the one millionth ; we should be content with finding it about 
382 thousandths. 

If the perpendicular distance between two successive leaves 
(called in botany an internode), is large, we can readily trace a 
helical line around the stem, passing in succession through the 
foot of every leaf. This maybe called the main or i^rincipal helix. 

and CA. we will put AC : CB = AB : BCA ; or "'-" — = — ^— . 

X 1 — X 

By proportion, we can add each denominator to its own 

1 — X 1 
numerator giving = ; and this, by transposing 

J) the members, is extreme and mean ratio, 1:1 — x = l — x:x. 

Again, that the quarters AD and AC should be as near 

equality as the thirds, we put AD : AC = AC : CB, or ^^--~ ""^ - 

1 — 2x xl X 

— - — ; and again adding the denominators ^--tt = 5 ^^^^^ adding the numera- 

X 1 X 

tors to the denominators, = — -^-^ which is extreme and mean ratio as before. 

1 — X 1 

The same result would follow at every step ; showing that this length of step would 

always make the nearest approach possible to equal division into as many parts as there 

had been points placed, whatever the number of steps. 




PHYLLOTAXIS. 81 

And it is evident that we may take the very same set of leaves 
and pass a helix in an oj)i:>osite direction through every leaf ; only 
in that case, the angular distance between two leaves will be 618 
instead of 382 thousandths of the way round. 

Let us now inquire, what further visible and easily observable 
evidence we should have, if a plant was actually constituted with 
this law of extreme and mean ratio, as the ideal plan of its distri- 
bution of leaves ; as seems to be demanded by perfect wdsdom and 
justice, and by symmetry. In case of the internodes being com- 
paratively short, this principal helix will wind round the stem 
with its threads so close, and the leaves so crowded that it will be 
difficult for the eye to follow the helix, or discover order in the 
arrangement of the leaves. Let us imagine the leaves on a piece 
of stem arranged in this ideal order, and numbered from zero, 
upward in the order of the stem. Split down the stem on tlie 
opposite side to the zero leaf, take off the bark, and spread it flat. 
If the internodes are short, the numbers over the zero will be 
arranged as those in figure A are arranged, when you turn the fig- 
ure cornerwise, so as to have at the bottom, and 55 almost over 
it, a shade to the right of tlie vertical. If tlie internodes are 
longer, the numbers are better represented by B. 

FIGURE A. 

15 23 31 30 47 55 
10 18 26 34 42 50 
5 13 21 29 87 45 
8 16 24 32 
. 3 11 19 
. . 6 14 



82 GEOMETRY AND FAITH. 

FIGURE B. 

15 23 31 39 47 55 
10 18 26 34 42 50 
5 13 21 29 37 
8 16 24 32 
3 11 19 27 
6 14 22 

In either figure the principal helix, joining consecutive num- 
bers, is transformed into a series of parallel straight lines. But 
we find also that straight lines from any selected leaf in any direc- 
tion, join equidistant numbers ; and every straight line, when the 
bark is rolled back into its cylindrical form, will become a helix. 
Thus the lines in figure A, joining 3 to 5, 6 to 8, and 10, 14 to 16 
and 18, etc., would become two secondary helices, steeper than 
the principal one. 

Again, it will beobserved that the numbers 0, 3, 6, etc., 5, 8, 11, 
14, etc., are in parallel straight lines ; and would make still steeper 
tertiary helices on the stem. 

Turning once more in the opposite direction, we find yet steeper 
parallel lines joining every fifth number; such as 0, 5, 10, etc.; 3, 
8, 13, etc. 

Still closer to the perpendicular, and in the direction of the 
original helix, we trace the lines 0, 8, 16 and 5, 13, 21, joining 
every eighth leaf. 

With a still greater compression of the length of the stem, and 
a diminution of the size of the numbers, lines would become con- 
spicuous, joining the leaves 0, 13, 26, etc.; or the leaves 0, 21, 42; 
with very great compression and very small numbers, even the 



PHYLLOTAXIS. 83 

lines 0, 34, 68, and 0, 55, 110, almost perpendicular, would become 
visible and conspicuous. 

Observe how curiously these numbers are generated from each 
other. Starting from zero on B, to go straight to 3 you pass be- 
tween 1 and 2 ; but 1 +2 = 3. Passing from to 5 we go close 
in between 2 and 3, and 2 + 3 = 5. Again the passage from to 
8 is between 5 and 3, and 5 + 3 = 8. The gateway from to 13 
is between 8 and 5 ; but 8 + 5 = 13. 

Observe, also, how readily these auxiliary helices will enable us 
to number the leaves. Suppose this piece of bark, with its foot- 
prints of the leaves on it, is before us, but the leaves not numbered. 
Assume any one as zero. Now here are three parallel helices 
taking in every leaf. Then upon the one passing through you 
can certainly write the numbers 0, 3, 6, etc. Here are five otlier 
helices crossing the first and including every leaf. Then upon the 
one passing through you can write 0, 5, 10, 15, etc. You can 
now pass to any leaf whatever by adding or subtracting 5 or 3, 
as you move up or down on either one of either set of helices. 

Were the stem crowded with a greater multitude of leaves, so 
that the more perpendicular and more numerous helices were most 
prominent, the same thing could be done. For example, were 
there 21 helices in one direction and 34 in tlie otlier, and you 
wished to find the 8th leaf from one which you had chosen as 
zero, you have simply to ascend two steps, on one of 21 leaf Hues, 
and descend one step on the 34 leaf line; since 42 — 34 = 8. 

It will at once be seen that if tlie stem of a ])lant be tougli, nnd 
of even texture, so that it will bear twisting evenly; tlien, if tlie 
leaves were arranged in this fashion, a twist in the stem would 
straighten up every alternate set of helices, and flatten down tlic 
others. A very gentle twist might, for example, bring the 341 h 
leaf over the zero. Every leaf, the twist being unif(M'm, would 
then be exactly over the 34th below it. The 34 leaf helices would 



84 geo:metry and faith. 

be vertical lines, and the leaves would be in 34 perpendicular 
rows. A little harder twist would brhig the 13th leaf over the 
zero ; and the leaves would then stand in 13 vertical rows. Twist 
still harder ; the 5th leaf comes over the zero ; the five helices be- 
come five vertical rows. If your stem is tough enough, twist vio- 
lently enough to bring 2 over the 0, and your leaves are in two 
rows, alternately opposite. 

Let us now imao-ine our p-entle twist at the beoinning^ to be in 
the opposite direction, and bring the 21st leaf over the zero ; this 
would give us 21 perpendicular rows ; a little more twist in that 
direction would dev^elop 8 rows ; a hard twist reduce them to 
three. 

This supposed twisting of the stem would increase or diminish 
the theoretically perfect angle, .381966 of the circumference, 
until, in the case of the two rows, the angle was .5 ; or, in the 
case of the three rows, it was .333333. Between these ex- 
triemes, ^ and Vs of the circumference, lie all those arithmetical 
approximations to the perfect angle, which we have developed by 
twisting; 1 : 2, 1 : 3, 2 : 5, 3 : 8, 5 : 13, 8 : 21, 13:34, 21:55,34:89, 
55 : 144, 89 : 233, 144 : 377. These approximate angles are formed 
each from the two preceding, by simply adding the two numera- 
tors, and adding the two denominators. 

One half is larger than .381966, and ys is smaller; and through- 
out the series they are alternately larger and smaller than the 
perfect angle. Xo other vulgar fraction stands in the line of 
direct advance. Take for example 7 : 19, it is not as near as 3 : 8, 
nor is 7 : 20, nor 5 : 14, nor any fraction with a denominator under 
21, except 5:13. This series settles toward the extreme and 
mean ratio like a vibratino; needle settlins; to the masrnetic merid- 
ian ; the last one given above differs from the true angle by only 
.001 of 1^ 

Suppose we should now take the case of an actual stem crowded 



PHYLLOTAXIS. 85 

with leaves, and should find that an actual twist to the left or to 
the right would bring the leaves into 3, 5, or 8 rows, what would 
it prove ? From the diagrams A and B it would prove that the 
leaves were actually arranged at equal angles around the stem, and 
that the angle was between 1 : 3 and 2:5 of a circumference. Sup- 
pose, however, that a slighter twist would bring the leaves into 13 
or 21 rows ; or suppose that the 34 and 55 leaf helices were already 
conspicuous, without any twist ; and suppose that by counting up 
by 34s and 55s we should find the 377tli leaf was, more exactly 
than any other, vertical over the zero leaf. This would prove the 
leaves to be actually arranged not only evenly around the stem, 
but at an angle almost precisely that of the theory. 

These suppositions are actually verified. Taking up, for ex- 
ample, a stalk of broad-leaved plantain, crowded with flower buds, 
or with seed pods ; you can, by twisting in one direction, bring 
out 8 or 3 rows ; by twisting in the other, 5 row^s, and with a hard 
twist, 2. Taking up any pine or spruce cone, and numbering a 
few scales by first counting the conspicuous parallel helices, in 
each of the two conspicuous intersecting sets, you will find the 
13th, 21st, 34th, or 55th leaf come most directly over the leaf 
taken as your zero. 

When the cylindrical stem is contracted into a strobile, cyme or 
rosette, or into the nearly flat head of a composite flower, the 
helices are transformed into other curves. Were the stem trans- 
parent, and the helices drawn in perspective upon a i)lane at right 
angles to it, the eye being the axis of the stem, tlie helices would 
become hyperbolic spirals. But the spirals of smiflower seeds 
make no a])proach to that form. Were a short ])()rtion of the 
stem, the height being equal to the radius, folded in toward the 
centre, the distance of the helix from the circumference being un- 
changed, tlie helix would become a s]ural of Conon, or Archimedes. 
But the sunflower does not conform to that law. 



86 GEOMETRY AND FAITH. 

The helices each make a constant angle with the meridian of 
the cylindrical stem; and as far as I have yet observed, this is 
true of the transformed curves ; so that on a globular strobile the 
helix becomes a rhumb line ; and on the sunflower head, a spira 
mirabilis of Bernouilli. In the rosettes of young JEnotheras, Cap- 
sellas, etc., the spiral is developed by a different process ; and may 
possibly be a different spiral, but the sunflower certainly approxi- 
mates (according to my measurement) very closely to a logarith- 
mic spiral ; the spira mirabilis. 

In the heads of composite flowers these spirals are beautifully 
conspicuous, and afford an easy method of determining the de- 
gree of approximation. The vertical lines have here become radii. 
There will always be two spirals (one running in each direction), 
plainly to be seen, crossing at each seed. The value of each of 
these can be determined by counting round the stem and seeing 
how many similar spirals there are in each direction. Thus, if 
there are 8 spirals, as it w^ere, parallel to each other, running round 
to the right, there they must be the 8 leaf helices ; and the 5 
leaf helices will be found running to the left, 5 in number. As- 
suming now any seed as a zero leaf, the adjacent seeds running 
up the spiral to the right will be the' 8th, 16th, 24th, etc. ; and 
running to the left the 5th, 10th, 15th, etc. By running to the 
left and right alternately you can thus determine the number of 
any seed as you please. By determining as nearly as possible the 
centre of the head, you can draw a radius, from your assumed 
zero and inward. If that radius, after leaving the zero, strikes 
first on the centre of number 34, the angle is 13 : 34 ; but if it 
steers between 34 and 55, grazes 89, and strikes a seed centrally 
first at 144, then the angle must be 55 : 144. 

As I was writing this, I picked up three heads of dandelion in 
seed, blew off the seeds, and counted the pits in the receptacle. 
One of them was on the 34 : 89 arrangement ; the other two each 



PHYLLOTAXIS. 87 

on the 55 : 144 arrangement. The largest of these was half an inch 
across ; and the error in the position of any two consecutive seeds 
in the outer row Avas, therefore, less than one 30,000th of an inch. 
One of the heads gave an angle a trifle too large ; the other two, 
an angle a little too small ; the average was almost exact. 

Subsequently, I counted the first ox-eye daisy, and the first sun- 
flower which I saw. They were equally exact as the dandelion, 
— the sunflower even nearer. It was upon a 144:377 approxi- 
mation; that is, the angle, instead of being 137°.508, was 137°.507, 
only about a thousandth of a degree too small. The head was 
about 20 centimetres or 8 inches in diameter ; so that two con- 
secutive seeds near the circumference would have an arc of 9^ 
inches or 24 centimetres between ; and this arc was actually about 
one 500th of a millimetre, less than the ten thousandtli of an inch, 
too small. It would surely be unreasonable to ask for any closer 
conformity of observation to theory. 

Nor is it easy to imagine any cause which necessitates the ar- 
rangement. It has been shown that if ^ cell generate cells on a 
horizontal plane at equal intervals of time ; and each cell begin to 
generate in the same manner, as soon as it is two intervals old : 
and the generation be always on a plane, and at the right liand 
side ; then the cells will be arranged like the seed pits in tlie dan- 
delion receptacle. But this goes very little way, — nay, I do not 
see that it even starts on the road, — toward showing us the gen- 
esis of the ascending helix. 

Again it has been suggested that as leaves grow by liu'lit and 
air, tliey will naturally grow where they have the best chance at 
getting them ; and this in the course of generations would lead 
them to come out exactly at the right spot. Unfortunntily for 
that explanation, the leaves of ))lants Avhicli need the light and 
air, and for whose benefit we liave invented tlie law, do not con- 
form to it in such wise as to make it a physical benefit. They 



88 GEOMETRY AND FAITH. 

grow almost universally on the lowest approximations, 2, 3, and 5 
ranked. 

Practically, the physical benefit is not felt. And in the part 
where the highest approximations are reached, in the heads, cones, 
cymes, etc., the physical benefit is lost through the crowding. We 
have evidently over-estimated, at the beginning of our speculation, 
the merely physical necessity of the arrangement. 

Of course we can set no bounds to the discovery of physical 
causes for physical effects ; and it is therefore possible that the 
botanist may, at some day, discover the physical agencies by 
which this physical arrangement of leaves is effected. But when 
he has done so, he will not have in the least shaken the theological 
inferences. The preponderance of the ruder approximations, 
1 : 2, 1 :3, 2 : 5, and 3:8, (1 : 3 and 3 : 8 giving too small, and 1:2 
and 2 : 5 too large, an angle) shows that the perfect phyllotactic 
law is not of practical importance in the growth of plants ; they 
live and flourish on the rudest approaches to it. But the tracing 
of these approximations up, in such very numerous instances, to 
the highest degree of accuracy, such as 55 : 44 and 34 : 89, one 
above, the other below the perfect, shows that the law of extreme 
and mean ratio is actually incorporated into the vegetable king- 
dom. The builder of the plant knew that law untold ages before 
the geometer invented it, to inscribe a pentagon. These succes- 
sive approximations point out more clearly and strikingly than 
absolute conformity to it could have done. 

And as its efliicient cause thus lay in the divine wisdom and 
divine power, so its final cause lies also in the spiritual realm. 
We have come upon it by an assumption that the leaves are 
treated justly ; that each is given the best possible chance at light 
and air. But while we have learned from our examination of it 
that the divine Architect knew this need of the leaf, and in pro- 
viding for it took this absolutely perfect law ; we learn also that 



PHYLLOTAXIS. 89 

he knew that perfect conformity was not physically needed ; and 
he therefore allowed these continued and great variations by 
which the law is suggested rather than thrust upon us ; he made 
the symmetry of the plant potential, rather than actual, and this 
suggestion, rather than actualization, of the perfect, makes the 
plant a more valuable teacher and companion of man. The sug- 
gestion of infinite perfection, that is beauty. 

The Lethe of nature 

Cannot trance him asrain, 
Whose soul sees the perfect, 

Which his eyes seek in vain. 

The outward eye cannot directly see the division of the circum- 
ference in extreme and mean ratio ; half the leaves are hidden, 
and even if we see two consecutive leaves, we cannot tell the 
precise angle. But the secondary helices of approximation are 
constantly visible, and give a great geometric fascination to the 
fructification, sometimes even to the foliage ; while in the larger 
growth of the plant the law secures general symmetry ; the varia- 
tion and concealment through various causes, prevent monotony 
and give an endless charm of variety. 



XV. 

NUMBER AND PROPORTION. 

It is only at a comparatively late jDeriod, in the development 
of the hmiian mind, that number comes into view, as a distinct 
object of thought. The idea of number is evolved from things 
of imperfect unity ; as an abstraction from things concrete, tan- 
gible and audible. The two hands, the ten fingers, the mother, 
the nurse, the window-panes, suggest the idea ; and it is slowly 
brought into the field of distinct intellection, by more or less 
laborious effort. A child usually attains the age of three or four 
years, before it gives evidence of attaching clear ideas to the 
names of numbers. Not until adult life do men usually perceive 
that persons are examj^les of the most complete and absolute 
unity. 

But this slowness with which the idea of number rises to the 
surface of consciousness, only shows how very deeply it is im- 
bedded in the soul. At the beginning of conscious life our atten- 
tion is fixed upon the individual ^objects presented in sensation. 
The child at that period, 

Nescio quid meditans nugarum : totus in illis : 

thinks only of the direct lessons of the outward world. The 
abstraction of number, and the invisible realities of space, of 
time, and of the spiritual world escape his attention, until he 
arrives at a mature condition. But the mature mind perceives 
that this historical order of the development of ideas is almost 
invariably precisely the reverse of the logical order of their de- 

90 



NUMBER AND PROPORTION. 91 

pendence. For example, space and time seem at first to be 
abstractions from the observed facts of matter and motion ; and 
it is hastily assumed that the experience of the outward world 
comes, at first, independently of any perception of space and 
time; and that these ideas are derived from that experience. 
This may be true in the chronological, or historical, succession of 
our distinct analytical attention to the ideas ; the actual sequence 
of distinct conscious attention is, that we first perceive motion, or 
rather matter in motion ; and this leads us to the consideration of 
space and time. 

Subsequent thought will, however, always show that, in the 
very first perception of matter in motion, we quietly take for 
granted the existence of space, occupied by matter ; and of time 
taken up in the motion. The ideas of matter, motion, space and 
time, actually enter the field of consciousness at the same instant ; 
that is, the historical order only relates to the sequence of the acts 
of attention, by which we separate the ideas from each other. 
In strict logical connection, the conception of space and time must 
precede the conceptions of matter and motion ; since space and 
time are the conditions on which motion is alone possible. 

In like manner it w^ill be found that although number is a late 
object of conscious attention ; and can be developed as an object of 
distinct consecutive thought only in a mind of some maturity ; it 
nevertheless stands logically antecedent to every act of intellec- 
tion. The conscious subject is conscious of an object ; and these 
with the act of consciousness form a tri-unity at tlie very begin- 
ning of any conscious life. If we dare venture so high a fiight of 
thought, we may even say that as the creative Mind must be pos- 
ited as a logical antecedent to creation ; so even in that infinite 
Mind, considered even as its own object, that same tri-unity ex- 
isted antecedent to any creation. Of course we sj)eak of logical, 
and not of chronological antecedence. Whether the latter ever 



92 GEOMETRY AND FAITH. 

really existed ; whether there was creation, in the widest sense of the 
word, is a matter beyond our most daring flight. But in the first 
act of intellection, the conception of subject and object implies 
the conception of number. As with the idea of self so with the 
idea of space and time ; they also necessarily imply number the 
moment that they are made objects of thought. Even infinite 
space, although absolutely homogeneous and without distinction 
of parts, except as such parts are created by thought, has its two 
elements of distance and direction ; and the element of direction 
has a manifoldness, which by no artifice of ingenuity can be re- 
duced to less than three dimensions. Some modern geometers, 
assuming that the conception of space is derived from, as well as 
suggested by experience, have speculated upon the possibility 
that, in those parts of the universe which are beyond the range 
of our experience, space may have other projDerties ; more dimen- 
sions than three, or a curvature by which two straight lines might 
include a surface ; and so on. Pursuing this speculation they 
have investigated certain algebraical sentences, expressing these 
impossible conceptions ; and have found the language capable of 
self-consistent interpretation. They have urged this possibility of 
self-consistent interpretation as an evidence that the conceptions 
themselves may be realized in the infinite distance. In spite of 
this ingenuity Reason sturdily maintains that the properties of 
space, as we see it here, are invariable throughout the whole ex- 
tent of absolutely boundless distances ; and although we may 
technically express, in algebraical language, the conception of a 
circle having unequal diameters, and deduce logically self-consist- 
ent results from the conception, yet we can neither make any 
picture of such a circle in our imagination, nor believe that such 
a circle can exist in any regions beyond the telescope. 

Number, inhering in tlie primal act of consciousness, follows 
every step of thought. All intellection, all thinking is the per- 



NUMBER AND PROPORTION. 93 

ception or creation of differences and distinctions, unities and 
resemblances. The definition of cliemistry, that it is the identifi- 
cation of the one in the many, and the detection of the many in 
the one, may be considered also as a definition of all science and 
of all thought. Number is more prominent in Chemistry, just as 
Space is more prominent in Mechanics, and Time in Biology ; but 
Number, Space and Time are all three involved in every finite act 
of intellection. All language bears witness to this presence of the 
three ideas in every thought ; take any word of any language 
and analyze it carefully, trace back its history and you find in it 
some more or less apparent reference to number and motion, taken 
perhaps as typical of spiritual things. 

But number, although thus involved in every act of conscious- 
ness, even the primal, is not the highest genus ; it is a species of 
relation. The highest unity is the person ; and the highest Per- 
son, although we speak of Ilim as Absolute and Unconditioned, 
stands logically related to Ilis own attributes and to His own cre- 
ation. Every act of finite intellection involves not only the per- 
ception of number, but of other relations also. There is more in 
the consciousness of subject, object, and relation between them, 
than the mere perception of tri-unity in the act. There is tlie 
perception of more than the numbers tliree, two, and one. Of 
course in the more complex acts of intellect, it is still more em- 
phatically true, that the ])erce])tion of numerical relation does not 
constitute the whole contents of consciousness. Number is, how- 
ever, the relation by which tlie relation of quantity becomes 
amenable to thought and calculation. 

At first, number is generated by the distinction of things dis- 
crete, and perliaps different in kind. It immediately becomes in 
itself an object of thouglit ; and its distinction of many or few 
1)eromes the most firmly gras])e<] and clearly comprehended of all 
relations of quantity ; which is at once ai)plied to the measure- 



94 GEOMETEY AND FAITH. 

ment of all things that may naturally be counted. But there are 
many kinds of quantity which are, absolutely or relatively, contin- 
uous ; and the measurement of the greater or less in these kinds 
is accomjDlished by the analogy of the greater or less quantity to 
large or small numbers. So indispensable to all clear intellection 
is this relation of numbers to each other, that the Greeks called it 
"^^0^0;^ that is word or Avisdom ; and the Latins called it ratio, or 
reason ; and this is its technical name among mathematicians to 
the present hour. The ratio of two numbers is their relation of 
magnitude, not as estimated by the excess of one over the other ; 
but estimated by how many times one is larger than the other. 
Thus the excess of 6 over 2 is the same as that of 8 over 4 ; but 
the ratio of 6 to 2, is 3 to 1 ; and that of 8 to 4 is but 2 to 1. 
When we seek to find the ratio of one continuous quantity to 
another of the same kind, we simply seek to find two numbers 
having the same ratio as the two quantities. One quantity is con- 
sidered as the unit, to which to refer the other. Usually as a pre- 
liminary step, both are first referred to some artificial unit ; such 
as an inch, a meter, a quart, a liter, a degree of the thermometer, 
a degree of angle, an hour, a dollar, etc., etc. 

Yet many of the most interesting ratios are found not to be 
equal to the ratio of any two numbers whatever. For example, 
the diagonal and side of a square, although nearly in the propor- 
tion of 17 to 12 are not exactly in the ratio of any two numbers 
whatever. And in general we may say that the mean proportional 
between 1 and another number, will very seldom be in a ratio ex- 
pressible in numbers. Two is a mean proportional between 1 and 4; 
because 1 is to 2, as 2 is to 4 ; 3 is a mean proportional between 1 
and 9 ; and so on. This mean proportional, or geometric mean, 
may be illustrated by letting fall a perpendicular from any point 
in a semi-circumference upon the diameter ; the length of the 
perpendicular is then a mean proportional between the two parts 
into which it divides the diameter. 



NUMBER AND PROPORTION. 95 

In the attempt to measure quantities, it is frequently necessary 
to assume a starting point, and measure in both directions ; — 
one is then able in subtracting to obtain quantities less than noth- 
ing; like south latitude, east longitude, temperature below zero, 
deficit in a treasury ; such quantities are called negative quanti- 
ties ; they lead us at once to the useful fiction of negative num- 
bers. But this fiction requires the additional fiction of negative 
ratios. For example, the ratio of 20 above zero to 10 below zero, 
is not simply that of 2 to 1, which would only give 10 above. It 
must be expressed by saying it is negative 2 ; — meaning that it 
is twice as far from the zero mark, but in the other direction. 

But out of this comes a very remarkable case of impossibility ; 
namely the impossibility of finding the geometric mean between 
a positive and a negative number. The geometric mean between 
1 and 2 cannot be expressed in numbers ; but the fraction |J is 
nearly it ; and no error can be named so small that a fraction can- 
not be named differing from the geometric mean less than that 
error. But the geometric mean between 1 above and 1 below 
zero cannot be expressed by any degree of the thermometer or by 
any conceivable numbers. It must be 1, but it can be neither pos- 
itive 1 nor negative 1 ; since 1 is not to positive 1, as positive 1 is 
to negative 1 ; neither is 1 to negative 1, as negative 1 is to nega- 
tive 1. Certainly zero is not the geometric mean ; for zero is not 
as many times negative 1, as 1 is times zero. What then is this 
temperature neither above, below, nor at zero? tliis latitude 
neither north nor south of tlie equator? It is called in mathemat- 
ics (luciis a non lucendo) tlie imaginary ; because it is unimagina- 
ble. The mathematicians have various symbols and various 
names for it, and use it freely in their calculations. They have 
endeavored, with great success, to reduce all quantitative and 
geometric impossibilities to this one ; that is to say, they Avill give 
you a correct answer to any absurd question you may devise, 



96 GEOMETRY AXD FAITH. 

providecl you allow them to introduce into their answers a sym- 
bol standing for the unimaginable mean proportional between pos- 
itive and negative unity. In geometrical questions this symbol 
may often be interpreted as signifying the rotation of a line about 
some point, but no general interpretation has been discovered. It 
is, however, a singular tribute to the ingenuity of the mathema- 
tician, that he has reduced all absurdities, all impossibilities, to 
this one, — the finding of an operation upon a quantity which 
shall neither increase it, diminish it, nor leave it unchanged. 
Give him a s}mibol, say /, for that operation, and he is as compe- 
tent to deal with the impossible, as with the possible. But the 
advantage and power, gained by the use of /, extends also into 
the realm of the possible and of the actual; and enables the math- 
ematician to assist in delicate and complicated researches of mod- 
ern physical science, otherwise beyond the reach of man. 

If we multiply 1 by 1.00000001, one hundred million times, the 
product is 2.7182818. It is the second of three famous quantities, 
pertaining to number alone, yet incapable of exact expression in 
number ; since exact expression would require the decimal part 
of the constant multiplier to be infinitely small, and the number 
of multiiDlications to be infinitely great. 

This peculiar number is known in mathematics by the name of the 
base ; (the base) ; and may be symbolized by h. It may be defined 
as the product of 100,000,000 factors, each equal to 1.00,000,001 ; 
but is susceptible of various other definitions. Like /, it furnishes 
the key to many physical problems, otherwise insoluble ; and it is 
very difiicult to express, in terms not technical, any reason why it 
should exert this beneficent power. 

The third of these famous quantities is the ratio of the circum- 
ference to its diameter ; which is nearly 22 to 7, still more nearly 
355 to 113 ; but which cannot be expressed exactly either in num- 
bers, or in mean proportionals between numbers ; it is a peculiar, 



NUMBER AND PROPORTION. 97 

unique ratio ; it may be symbolized, if you please, by c ; and its 
immense value in calculation arises partly from the fact that the 
difficulty of measuring not only circles, but every conceivable 
kind of curved lines, surfaces and solids, is generally reducible to 
the calculations of b and c. Add i and you have the means of 
calculating the inconceivable also. 

These three famous ratios, so entirely different in their origin, 
and so utterly incapable of exact expression in number, may be 
connected by a very simple bond. If we multiply 1 by 2 ten 
times we obtain 1024 ; and if we multiply 1 by 1024 we obtain 
the same. Since ten equal multiplications by 2 are equal to one 
multiplication by 1024 ; we may say that multiplying by 2 is mul- 
tiplying by 1024 one-tenth of a time ; multiplying by 8 is multi- 
plying by 1024 three-tenths of a time, and so on. Using the like ex- 
pression we may say that b multiplied by itself half c times gives 
4.8148. Divide 1 by this number and we get .20788. Next we 
take the impossible, inconceivable, inexpressible i, and multiply it 
by itself i times ; and can readily demonstrate that it produces 
identically that same .20788. Here then are the three famous and 
inexpressible ratios of numbers, — base, circumference, and the 
imaginary, — bound together in this simple law ; to wit : Multiply 
base by itself, half circumference times ; then multiply the pro- 
duct by imaginary, imaginary times; and the final product is 
unity. 

The object of this long discussion of such abstruse mmicrical 
relations is to set in a more striking light the marvellous ])ower of 
the human body as an unconscious calculating engine, alluded to 
in more than one of the preceding chapters. Number, the crea- 
tion of conscious spirit, created in the act of consciousness, cannot, 
in its utmost reaches of abstruseness, go beyond the ])ower of the 
Infinite Mind. But in our finite thoughts we reach conclusions, 
like the above, binding together what is conceivable, but inex- 



98 GEOMETEY AND FAITH. 

plicable, with what is neither conceivable nor expressible, in one 
inexplicable bond. 

Yet feeling, or emotion, is a higher, deeper state of conscious- 
ness than thought; and often carries us into regions of the 
Divine thought where finite thought is unable to follow. In some 
instances the reality of this flight into higher regions has afterward 
been verified, by the slower laborious ascent of the finite thought, 
in paths over which feeling had flown. Thus early thinkers de- 
clar.d that in the perception of musical harmony, the soul un- 
known to herself calculated secretly the numerical ratios of the 
undulations of the air. Modern musical statics has proved that 
these secret, unconscious numerical calculations by the musical 
ear, by the mere aesthetic feelings, have been far more numerous, 
complicated and precise than the older thinkers had supposed ; 
and that elaborate comparison, calculation and experiment demon- 
strate the course and progress of music, from barbaric times to 
the present, to have been in perfect agreement in all respects 
with laws of number, not revealed until very recently. It is 
certainly difiicult to imagine modes by w^hich experience and 
habit could have led to this progress, had not the human body 
been in its very creation formed in exquisite adaptation to those 
laws. 

I have, in another chapter, alluded to Hay's law that beauty of 
geometric form depends upon the division of the right angle in 
harmonic ratio. Burke argues with a great deal of misplaced in- 
genuity to prove that proportion has nothing to do with the cause 
of beauty ; but Hay simply takes the Parthenon, the Temple of 
Theseus, the Lincoln Cathedral, and other universally recognized 
types of beauty, and shows that their actual and potential angles do 
always stand in simple harmonic ratio to the right angle. Here, 
therefore, is a second instance in which the secret unconscious 
calculations of the beholders have always recognized by feeling 



NUMBER AND PROPOHTION. 99 

what the labors of Hay have shown to be numerical proportions 
in architecture. 

But I think I have proved that Hay's law is incomplete ; and 
that the eye for beauty recognizes sometimes a far more delicate 
and involved j)roportion than that which he assumes. He illus- 
trates proportion of geometric stationary angle by the proportion 
of vibrations in time, producing a musical concord ; and draws his 
law from an examination of architectural and plastic art. But the 
beauty of the vegetable world led me to consider that the phyllo- 
tactic ratio fully developed in the preceding chapter might also 
give beauty in artificial forms. The approximations J, -J, |, and f , 
would be also included in Hay's series. To test this question I 
have tried various experiments with satisfactory results. A single 
one will show their nature. I drew two semi-ellipses ; in one, the 
angle (from the end to the middle, and to the mid side) was f ; so 
that either by phyllotaxy or by harmony it should be beautiful. 
The other was drawn exactly on the perfect phyllotactic angle ; 
which, as harmony, would be rudely discordant. Showing these 
two ellipses, without explanation, to many persons, in private and 
in schools, and taking a vote on the merits of their shape, the vote 
was unanimous on their both being good, very many preferred 
the phyllotactic angle. They were so nearly alike in their propor- 
tions that some observers mistook, in attempted conscious calcula- 
tion, and thought the higher one flatter, even although joining in 
the judgment that it was more beautiful. 

How much more wonderful, if possible, is that unconscious cal- 
culation of numerical relations not yet confirmed by figures, but 
proved by experiment to be correctly performed in higher de])art- 
ments of art. The harmony of colors unquestionably depends on 
numerical relations of wave lengths. The workman at a Bessemer 
steel factory knows when to draw the charge by a secret uncon- 
scious calculation of the rapidity of vibrations, millions in a 



100 GEOMETRY AXD FAITH. 

second, taking place in the flame. All the world have by similar, 
but more delicate computations, confirmed and endorsed Titian's 
judgment in the harmony of colors. What is a great painting 
but the conveyance of great thoughts and feelings from the painter 
to the beholder through medium of numerical ratios of angles in 
the drawing and of vibrations in the coloring. It is ratio ; it is 
the logos, the incarnate word of the painter, imitating the Al- 
mighty Logos by which the heavens and the earth were made, 
and by which the painter's hand had received the skill to make, 
the beholder's eye the skill to interpret, the imitation. 

The same questions may be asked, the same answer must be 
given, concerning the higher ends of music. I have demonstrated 
by hundreds of carefully conducted experiments ujDon hundreds 
of persons, sometimes upon large classes in schools, that fully 
three-fourths of an audience receive from a musical composition 
the very same moral mood or tone of feeling which filled the com- 
poser at the time of composing it. The closing chorus in Beetho- 
ven's " Mount of Olives," played upon an organ, without words, 
gives to a hearer who has not heard it before, and who is not in- 
formed of what it is, immediately and irresistibly an impression 
of solemn awe, of inexpressible majesty, of penitence, yet of the 
peace of forgiveness ; of gratitude for forgiveness, and a sense of 
reconciliation through mediation. All this is accomplished sim- 
ply through rhythmic modulations ; but not, as in spoken words, 
through any conventional or artificial associations with them. It 
is the direct natural language in which Beethoven utters his pro- 
foundest, deepest faith. If the hearer have known anything con- 
cerning the tenets of the Christian faith, he will be certain that 
the composer was a Christian believer, uttering through those 
chords a thanksgiving, in the name of all worlds, for the recon- 
ciliation of the world through the cross. What explanation can 
be given of this high power of the artist, in whatever department. 



NUMBER AND PROPORTION. 101 

to pierce direct to the heart without the aid of conscious intellec- 
tion in the head ? I see none, except to admit that given in 
Emerson's " Problem," namely, to admit that the forces of nature, 
partially under our conscious guidance, are wholly under the con- 
trol of that same infinite wisdom and love which inspires the 
artist, and interprets his work to his audience. 



XVI. 

THE DEVELOPMENT OF FORMS. 

It is sometimes said that classification always proceeds by a 
process of dismissing from attention that which is peculiar to the 
individuals ; and fastening the mind upon some arbitrarily chosen 
points of resemblance. Examples of this method are found in 
Linne's artificial system of botany, and in Agassiz's proposed 
classification of fish by their scales. All classification does not, 
however, proceed upon this method. Again, it is said that the 
classification of the organic kingdoms is an arrangement accord- 
ing to the degree of development ; that development has been a 
continuous process ; and the degrees have been accidentally, or 
else arbitrarily marked out< A little examination Avill show that 
this view, also, is erroneous. 

The classification of the organic kingdoms is unquestionably 
based upon form, in the geometric sense of the word. An accurate 
drawing, an accurate plaster-cast, the impression left in a rock, — 
these are sufficient data for a naturalist to decide the identity of a 
species. If, therefore, classification is built upon development, it 
is upon the development of forms in space. All these forms have 
a certain amount of symmetry ; their outlines and surfaces corre- 
spond, more or less perfectly, to the geometric law. VTe have 
already shown that this conformity to law is obedience to thought. 
But not only does the form of each individual conform to law ; 
the series of forms, also, as Agassiz has shown, indicates a law 
pervading the series ; and this fact is inconsistent with the opin- 
ion that evolution, if there has been evolution, has proceeded con- 
tinuously, by insensible gradations. 
102 



THE DEVELOPMENT OF FORMS. 103 

All scientific men, at the present day, admit the reign of law in 
organic nature ; even those who believe in continuous develop- 
ment. The origin of the universal, invariable law of inorganic 
nature cannot, however, be discovered by an investigation con- 
fined to nature herself. That question lies outside the realm of 
science. The scientific man may consider it, and may answer it, 
wisely or unwisely ; but in thus doing he has left the domain of 
science, and entered that of philosophy. Science infers the uni- 
versality and invariability of natural law, by a '' simple enumer- 
ation " of the increasing number of observed phenomena con- 
forming to law; philosophy takes a firmer ground. She con- 
ceives the universality of law to flow as a necessary consequence 
from the grandest and most certain conclusion of human thought ; 
the being of an infinitely wise Creator. The infinite Wisdom 
foresaw from eternity the best possible modes of action, and 
adopted them. No occasion can arise to make Infinite Wisdom 
change its plan. " By the determinate counsel of the Lord," says 
the wise Hebrew, " were his works from the beginning ; " " He 
gave eternal order to his works, and to the atoms thereof, for all 
generations ; " " nor to eternity shall they disobey His word." No 
scientific writer of the nineteenth century can more distinctly 
afiirm the universality and invariability of law than this religious 
teacher more than twenty centuries ago. Historically, as well as 
philosophically, the doctrine is the direct outcome of theistic faith. 
To one whose mind has grasped the conception of the existence 
of an infinite Deity, of infinite wisdom and j^ower, tlie whole 
universe becomes the expression of a single Divine Thought, 
almost infinitely full of detail, and thus giving endless occupation 
to our finite intellect ; but also possessing perfect unity, and thus 
flooding us with the fulness of its beauty. In this theological 
form the doctrine of the correlation of forces, and interdependence 
of all sciences, was familiar to metaphysical and theological writ- 



104 GEOMETRY AND FAITH. 

ers, long before its tardy confirmation by the induction of 
physicists. The natural sciences, like the mathematics, have their 
postulates ; among them is that of the invariability of law ; they 
need also the postulate of the universality of law. The doctrine 
of a continuous development will be found difficult to reconcile 
with either. 

This doctrine is a virtual denial of the existence of law in a 
department in which the whole history of science would lead us 
most surely to expect the discovery of law. Evolution itself has 
strong antecedent probabilities in its favor ; it is only against the 
mode of evolution by continuous change and accidental arrest, at 
certain accidentally determined stages, that the present objections 
lie. The whole course of scientific study up to the middle of this 
century had revealed more and more clearly the j^resence of law, 
governing the atoms, and molecules, and masses of the inorganic 
world. In order to do this, it had called in the aid of the mathe- 
matician. By his technical language, alone, can any problem of 
time and space, matter and motion be exactly and definitely 
solved. Even in the organic world, he had begun to render 
the botanist and zoologist important services. He had shown 
that, in the plant, there is a law of extreme and mean ratio ; that 
this is the nearest approach to a uniform distribution of the leaves ; 
he had shown that, in the zoologist's classification of animals, the 
first great division rests upon laws of mechanical equilibrium in 
the embryo, as imperious as the law of an arch. The natural ex- 
pectation was that mathematical bases would in like manner be 
found for the division of classes, orders, families and genera, if not 
the species. 

But under the theory of continuous gradual development this 
course of scientific progress seems arrested. From the fact that 
between any two forms of nature we can always, if we have 
numerous specimens, find intervening links, it is assumed that 



THE DEVELOPMENT OF FOKMS. 105 

there is no real line of demarcation between the forms. There is 
a double fallacy in this assumption. Between any chestnut and 
any oak, for example, it may be possible to find an intermediate 
form. But it does not follow that it would be possible to find a 
form of which a thoroughly trained botanist would say, " This is 
either an oak or a chestnut, but I cannot tell which." It may 
easily be true that no definition, in words, of an oak would ex- 
clude every chestnut ; nor any definition of a chestnut exclude 
every oak. It does not follow that the senses would not distin- 
guish; a sharp observer sees many differences not easily to be 
described in words ; as those between an oak and a chestnut, an 
apple and a pear, a plum and a cherry. The history of science is 
full of instances in which a more acute observer has learned to see 
and point out distinctions between things which had been con- 
founded. And, secondly, if it were possible to show that between 
two groups the transition is so gradual that no eye can detect the 
dividing line, it w^ould not follow that no dividing line existed, 
nor that the groups had a common origin. The intellect can some- 
times clearly distinguish things, indistinguishable by sense. The 
elastic curve has, at the extremities of its series of variations, a 
straight line and a circle ; indistinguishable by the eye, or by the 
imagination, from those produced by varying the eccentricity of 
the ellipse. Yet reason, transcending imagination, shows tliem to 
be entirely different ; they cannot be made alike, except by a 
process which would confound all intellectual distinctions, and de- 
stroy the possibility of science. 

The ellipse and the elastic curve belong to genera further re- 
moved from each other than the oak and the chestnut ; yet each 
may pass by variation into the form of a circle. Similar instances 
abound in the curves investigated by geometry. An observer 
unacquainted with mathematics miglit tliink it easy to pass tlie 
elastic curve into the form of a circle, and then elongate it to an 



106 GEOMETRY AND FAITH. 

ellipse ; or easy to frame a definition of the one curve, which should 
also include the other ; the geometer knows that neither is possi- 
ble. When the forms of the chestnut and the oak are as thor- 
oughly understood by the botanist, as those of the ellipse and the 
elastic curve are by the geometer, he will probably Avonder that 
the tAvo genera were ever considered difficult to define and separate. 
It may even be that the mathematician will demonstrate the dif- 
ference of the forms. It is the mathematician alone who has in- 
troduced precision and certainty into the other physical sciences ; 
and he will probably, at some day, introduce them into biology. 
The botanist and the zoologist may rebel, but they will rebel in 
vain. The numbers of Pythagoras and the axioms of Euclid are 
inexorable. The fates themselves cannot violate the laws of geom- 
etry, arithmetic, and algebra ; much less can fluttering theorists 
break through those adamantine bars. 

The erratic genius of De Maillet, and of Erasmus Darwin, has 
built an ingenious theory, perfected under the magic influence of 
Charles Darwdn, by the aid of a host of writers, which flatters 
itself that it explains all organic forms in complete independence 
of geometrical and arithmetical law. It makes the labors of 
classification as empty of real meaning as though they had been 
expended upon the forms of clouds or upon the disposition of the 
settlings of cups of tea. This theory of insensible, accidental 
variation, modified and arrested by the surroundings, is a virtual 
assertion that the whole problem of classification is a delusion. 
Expand the theory to its most complete form, and it becomes self- 
contradictory and self -destructive. It would declare all personifi- 
cation of objects, and all entification of attributes, mythical and 
illusory, and make monotheism the last step of the illusion pre- 
vious to awaking and discovering that it is all a dream. Thus its 
evolution contains in itself an illogical breach of continuity. 
When a mathematical series tends through an indefinite series of 



THE DEVELOPMENT OF FORMS. 107 

terms toward unity, it is illogical to assume either that it ends in 
zero or that it becomes indefinite. As well might the modern 
correlation of forces, tending constantly to show that all phenom- 
ena are modes of motion, be held to show that there is no motion, 
or at least that we can not know that there is motion, as the 
universal tendency of cultivated thought to reduce all super- 
natural forces to the will of one God be held to lead properly to 
atheism or to agnosticism. If the reduction of personification in 
things and ideas to smaller and smaller numbers logically leads 
to the denial of real being in any entified idea, it should lead 
also to the denial of personality in any thing; that is, to the 
denial of personality in our fellow-animals, and even in our fel- 
low-men. In fact, this theory goes further; it ignores the ex- 
stence of personality in one's self ; it makes the theorist himself 
a non-existent being, knowing only that one thing, that he does 
not exist. 

The students of botany and zoology have been laboring for 
nearly twenty-five centuries in one direction, with steady pro- 
gress ; there has been a substantial agreement among them as to 
the proper divisions of the animal and vegetable kingdoms ; their 
divisions have been concerning the grounds of the division. For 
the final settlement of the questions of classification the aid of 
the mathematics is necessary. This has been tlie destiny of 
the other physical sciences ; it is tliat of biology also. There is no 
breach of continuity in nature ; geometric and algebraic law rules 
absolute over all that can be bounded in space and time. The 
vagueness of arbitrary variation and survival of tlie fittest is a 
poetical dream ; it must give way to the intellectual, scientific 
sternness of invariable law, bounded by invariable limits. As the 
four primitive forms of the embryo flow from necessary mechani- 
cal conditions, inflexible as tlie law of equilibrium in arches ; so 
the classes, orders, families, genera, will be found to be fornieJ 



108 GEOMETRY AND FAITH. 

by conditions sharply defined in nature ; and hereafter to be, 
through the aid of mathematics, sharply defined in human 
thought. 

No geometer will willingly relinquish his hope of great triumphs 
in the future ; when the new mathematical methods of the nine- 
teenth century shall have been as faithfully applied to the prob- 
lems of organic form, as the methods of the seventeenth century 
have been to those of inorganic matter. If the melodious wail of 
Darwin's bugle leads the naturalists to retreat from the grand 
problem of classification ; and if the trumpet of the elder Agassiz 
fails to rally them ; the mathematicians will press forward and 
gain the high honor of victory, on the noblest field of natural sci- 
ence. Mathematical science cannot admit the possibility that the 
rhythm and symmetry of the organic kingdoms is an accidental 
result of accidental variations ; there must be algebraic and geo- 
metric law at the basis, not only of each organic form, but of the 
series of forms. The series has a unity ; capable, when men have 
attained a fuller comprehension of it, of expression in terms of 
thought. 

The rhythm and harmony of a symphony reveal not only the 
skill of the orchestra and its conductor; but the great mind and 
noble heart of the composer. The rhythm and harmony of the 
organic world reveal the power, the wisdom, and the love of God. 
So long as man is less than the universe, his wisest and best course 
is to seek everywhere, not for discords and maladaptations, but for 
harmonies, correlations, adaptations. The universe is the sum of 
all symmetries ; and contains all geometries, architectures, sculpt- 
ures, and pictorial arts. It is the sum of all rhythms, melodic or 
harmonic ; and contains all algebra, poetry, music and dance. 
The Divine Word, which created it, is wisdom and love ; and 
manifests wisdom and love in every syllable and tone in which 
it utters itself; not least in the wondrous series of the forms 



THE DEVELOPMENT OF FORMS. 109 

of plants and animals; swaying, in the responsive rhythm of 
growth and decay, sleep and activity, generation and succession, 
to the periodic march of the earth, the moon, the planets and 
the sun. 



